Broader View of Relativity: General Implications of Lorentz and Poincare Invariance

where a/aw and HQ both transform as the zeroth component of pi1. Since the taiji-time w has the dimension of length, Hn. has dimensions of inverse l e n g t h and the Dirac equation leads to electronic energy levels also with dimensions of inverse length. Solving (10.2) gives the electronic energy levels for hydrogen atom

an

2

(10.3) 1+n - (j + f) + j j

+ f)2-«e

where a e is the fine structure constant a e = e 2 / 4 i t = 1/137.036.

In

natural

units, the electron mass m e is 2.59..xW u cm 1 . Thus in taiji relativity, an atomic system can only emit or absorb energy q u a n t a characterized by a 'taiji frequency'

k° (with dimensions of inverse length) that is equal to the

difference between two energy levels k°= IE n - E kilt is important to note that in taiji relativity, where we have only a unit for length, the traditional concept of frequency (co or v) is not defined.

Instead,

an electron making a transition from a higher to a lower energy level emits radiation that travels with a speed PL=1 and that has a wavelength X=2ji/k w h e r e k=lkl = k° is the difference in energy between the two levels. This wavelength X

Experimental Tests II

129

can be measured experimentally by passing the radiation through a diffraction grating and measuring the positions of the light and dark bands that result from the interference.

Although we will call k° the taiji frequency because it is

analogous to the frequency in special relativity, it is more closely related to an energy in taiji relativity. Because the invariance of the energy-momentum law implies that the quantity k^k^ = (ko) 2 - k 2 = 0 is invariant, the wave 4-vector kV- must transform in the same way as the coordinate 4-vector xV k' 0 = Y (k°-|3k x ),

k' x = Y (k x -pk°),

k' y = k y ,

k' z = k z .

(10.4)

Since kx=lklcos8=k°cos8,

(10.5)

the Doppler effect in taiji relativity is k , 0 = k°Y(l-(3cose).

(10.6)

The second order term of the transverse Doppler shift (when 8 = 90°) is thus AX = (1/2) p2X., the same as in special relativity.3

10c.

Atomic energy levels and Doppler shifts in common

relativity

In common relativity, where we use the more familiar set of units (with the exception of the replacement of the second by the common-second), the covariant Dirac Hamiltonian for a hydrogen atom in the F frame is e2 HD=-aD-p-pDm--—, 4nr

p = -UV,

e = -1.602xl0- 2 0 -\/4^(g-cm) 1 / 2 ,

130

J = 3.5177293xl0-38gcm ,

A Broader View of Relativity

(10.7)

where e has the numerical value e/c and J has the numerical value h/c.

The

covariant Dirac equation in a general inertial frame is

U-T^-VT-HDV,

(10.8)

a(btc) where both a/a(bt c ) and HQ transform as the zeroth component of a 4-vector. Since the quantities Ho and J(a/fl(bt c )) both have the dimension of mass, the Dirac equation leads to electronic energy levels with the dimension of mass. Solving the Dirac equation (10.8) leads to 2

(10.9)

n-(j + i) + J(j + i) 2 -

again, where oce= e 2 /(4nj) = 1/137.036 is the fine structure constant.

Thus in

common relativity, we can think of an atomic system as having "mass levels" M n and such systems can only emit or absorb "mass quanta" that are equal to the difference between two mass levels. These quanta, with a value Jk° = IM„-M k l,

(10.10)

are directly proportional to the zeroth component of the wave 4-vector k°. Since common relativity has a separate unit of time, we can define a frequency oo so that the wave 4-vector can be written as

kM-,k

in F

and

k'^= — ,k'| in P .

(10.11)

To see an important difference between common relativity and special relativity regarding frequency and the speed of light, consider first an atom at rest in the F frame in which the speed of light is isotropic. As measured by an F observer,

Experimental Tests II

131

an electron making a transition between two mass levels M n and Mk will radiate a photon with a frequency
(10.12)

In contrast, consider the same kind of atom at rest in the F' frame, where the speed of light is not isotropic.

For measurements made by an F' observer, we

have |Mt n - Mtkl= S'oi'/ct 2 =(/jVd) kto,

(10.13)

where we have used the t symbol to indicate that these are quantities for a different physical system (one at rest relative to F', rather than relative to F). 4 Since the atoms are at rest in F and F' respectively, the principle of relativity requires that (M n -Mk) and (M f n - M\) both be equal and isotropic as measured by observers at rest relative to those respective systems. Thus, we get (/i/OkOM/iVcWO,

(10.14)

because all inertial frames are equivalent. Since the principle of relativity also demands that coVc1 = co/c, or equivalently that k t0 =k°, t h e n 4 S/c = ^ / c t = J .

(10.15)

Thus, the ratio Ti/z is a universal constant in common relativity even though the values of h and c, separately, are not.

Because in the equations of common

relativity, this ratio appears everywhere in place of fi, we will denote it by the letter J. This concept of the universal constant J in common relativity and the relations (10.9)-(10.15) are very important to understanding the experimental results of Doppler shifts within the framework of common relativity. Within the framework of common relativity, it is not the shift in the frequency w that is measured in experiments, but the quantities co'/c' =k'° and

132

A Broader View of Relativity

co/c=k°.

Therefore, the Doppler shift

is given

by the

four-dimensional

transformation of the wave vectors k^ and k'V (in F and F' frames):

k'° = Y (k°-pk x ),

k'x = Y (k x -pk°),

k'y = k y ,

k'z=kz.

(10.16)

This satisfies the invariance relation, (k'°) 2 -k' 2 =(k°) 2 -k 2 =0,

(10.17)

for electromagnetic waves. Since kx=lklcose=k°cose, we obtain the same Doppler shift as in special relativity 3

k'° = k°l-pcose

(10.18)

fV

consistent with the measurements obtained by Ives and Stilwell. 1 lOd. Lifetime

dilation

of

cosmic-ray

muons

The final experiment that we consider from the point of view of taiji and common relativity is the dilation of the lifetime of unstable muons produced in the upper atmosphere by cosmic rays. In 1941, B. Rossi and D. B. Hall published the results of a study in which they investigated the flux of muons with energies within a certain range at various altitudes. 5 The flux of muons with a particular range of energies is measured on a mountain top, a few kilometers above sea level.

Since these muons travel at nearly the speed of light, the

amount of time required for the muons to reach sea level can be computed and that number compared with the decay rate of muons at rest to obtain the expected muon flux at sea level. However, measurements made at sea level show a much larger flux of muons than expected from the calculations. It is as if, to the muons, the trip took much less time t h a n

was calculated.

These

measurements have been interpreted as confirmation of the relativistic time dilation effects resulting from the particular clock synchronization

Experimental Tests II

procedure

133

of special relativity. Can the results of this experiment possibly be predicted by common relativity, in which a different clock synchronization procedure is used? As we shall see, the answer is yes. lOe. The cosmic-ray muon experiment

and taiji

relativity

As with the other experiments, because the spacetime transformations in taiji relativity are mathematically identical to special relativity, it is clear that taiji relativity must predict the correct outcome for this experiment.

Again,

however, we analyze this experiment in terms of natural units to expose what is truly revealed by the experimental results and to set the stage for how common relativity with its clock synchronization procedure that differs from the one in special relativity, can produce the same predictions as special relativity. The cosmic ray muon experiment can be analyzed in two ways, one i n terms of a simple dilation of the "decay length" of the muons and the second in terms of the transformation of transition probabilities as calculated by quantum field theory in general. We begin with the former analysis. We consider an unstable particle such as a muon at rest in the F' frame (Ax'=x'2-x'i = 0) and moving with a velocity p as measured in the F frame.

If the

time interval between the creation and decay of the particle is Aw'0 as measured by an F' observer, then the corresponding interval Aw as measured by an observer in F is AW = YAW'0,

(10.19)

the same result as in special relativity. Because the unit second is not defined in taiji relativity, we can think of this result as a dilation of the "decay length" of the particle, where "decay length" means "the vacuum optical path length traversed by a light signal that travels a closed path in F'(Ax'=0) between the time of creation and the time of decay of the particle." Such an optical length helps us understand the concept of "rest decay length" Aw'0 for a particle decay at rest in taiji relativity.

134

A Broader View of Relativity

lOf.

Decay-length dilation in quantum field theory and taiji relativity

We now turn our attention to an analysis of the muon decay from the point of view of transition probabilities in quantum field theory (see Appendix C). In the covariant formalism of quantum field theory based on taiji relativity, one starts with an invariant action and uses the taiji time w in a general inertial frame as the evolution variable for a state in the Schrodinger representation. Because the evolution of a physical system is assumed to be described by a Hamiltonian operator H^s*(w) with the same transformation property as the taiji-time w or the operator a/aw, the equation of a physical state *( s )(w) in the Schrodinger representation is given by

ii*^vO-H(S)(w)*(S)(w)i AW!

H(S) = H ( S ) + H T ( S ). O

(10.20)

1

The usual covariant formalism of perturbation theory can also be applied to quantum field theory by considering the interaction representation and the S-matrix based on taiji relativity. To see the dilation of the mean decay length of an unstable particle, let us examine the decay rate r(l-»2+3+...+N) for a physical process l-»2+3+...+N. This rate is given by r(l-*2+3+...+N) = »*„ f i-l
d3xNd3pN _ (2jt)3

(la21)

which has the dimensions of inverse length in taiji relativity. Since a particle's lifetime is measured in terms of taiji-time, which has the units of length, this decay rate can be interpreted as the inverse of the particle's "decay-length" or the mean distance which a group of such particles would travel before their number is decreased by a factor of l/e=l/2.71828 due to decay. The decay length D is given by D = l/T(l-2+3+...+N) .

(10.22)

Experimental Tests II

135

Thus, in taiji relativity, one has the "rest decay length" D 0 for a particle decay at rest, corresponding to the "rest lifetime" in the conventional theory.

Also,

instead of the dilation of the lifetime of a particle in flight, we have dilation of the distance it travels before decaying.

Such a dilation is physically correct

because it is equal to the experimentally determined distance traveled by the particles at high energies and at a speed very close to the speed of light. 5 Let us consider a simple specific example, i.e., the muon decay u~(pi) -* e"(p2)+v (i (p3)+v e (p4) with the usual V-A coupling. The decay S-matrix in the momentum space is given by (fISli) « ( p 0 1 p 0 2 p 0 3 p 0 4 ) - 1 8 4 ( p 1 - p 2 - p 3 - p 4 ) M s c ,

(10.23)

M sc = (G/V2 )[vn(p3)YX(l-Y5)n(Pi)][e(p2)YA(l-Y5)Ve(P4)] • The decay length, defined by D=l/r(l->2+3+...+N), is found to be

I

D

d3p2 d 3 p3 d 3 p4, 4 6 (Pl-P2-P3-P4) 2 I M s d 2 " P01 P02 P()3 P()4

1

(10.24)

spin

Everything to the right of 1/poi in (10.24) is invariant under a taiji transformation

so t h a t the

decay length

D is

indeed proportional

to

P01 = m ^ = nij / T/1 - p? .5 We note that, when a particle at rest decays, its decay-length is not zero and can be expressed in terms of taiji-time w. It should be stressed that result

(10.24) is sufficient

to u n d e r s t a n d all

previous

experiments of the "lifetime dilation" of unstable particles because it is the decay-length of an unstable particle in flight which is the quantity that is directly measured in these experiments.

For example, in the

experiment

described previously involving the counting of the muon flux at various altitudes above sea level, one is actually measuring the distance a group of muons travels in order for their number to be reduced by a certain factor and one then infers information about decay times from combining the distance information with calculated velocities of the muons. 6

136

A Broader View of Relativity

Similarly, the experimental setup in laboratories for measuring "lifetime dilation" is roughly as follows:

SI

S2

S4

S3

Fig. 10.1 Experimental arrangement for measuring the mean lifetime of particle decays in flight. A narrow beam of unstable particles, for example, pions, passes through t h r e e detectors, Si, S2 and S3. Some of the pions decay between detectors S3 and S4, according to the reaction ji + ^-n + +v ( i .

If the distance L between S3 and S4 is

sufficiently large, the probability of a \i+ entering the detector S4 is negligible because the n+'s produced in the decay move out in all directions.

The

coincidence of signals in Si, S2 and S3 indicates a high-speed charged particle passing all three detectors, i.e., moving in the horizontal beam direction.

By

counting these coincident signals, one has the total number of charged pions N0 in the beam before decay. Similarly, the coincidence of signals in S i ^ ^ and S4 gives the number of pions, N' remaining after traveling a length L in the horizontal direction of the beam. In taiji relativity, the relationship between No and N' can be given in terms of taiji-time as,

N' = N 0 exp[-rw'],

r = r ( ™ + - n + + v ( l ),

(10.25)

where w' is the proper taiji-time of the pion (at rest in the F' frame, i.e., Ax'=0) moving from S3 to S4 . This relationship can also be expressed in terms of the taiji-time w of the laboratory frame F, 2 w = wVl-P

(10.26)

Experimental Tests II

137

where B=dx/dw is the known taiji-velocity of the pion.

Thus, using w=L/B,

equation (10.25) can be written as N' = N0 exp[-rwVHJ2 ] = N 0 expf-lVlHp L/B] .

(10.27)

Experimentally, one changes L and measures N' and N0 for each L and then plots a graph of ln(N'/N 0 ) versus L to get a straight line with a slope of -rVl-B 2 /B. From the slope and the known taiji-velocity B, one can then obtain the mean life of a positive pion, W(JI+) = D(JT+) = l / r = 780.45 cm .

(10.28)

Since the time interval At=l second corresponds to Aw=299792458 meters, t h e result (10.28) corresponds to the conventional pion mean life T(JI + )=2.603X10- 8 seconds in special relativity. Within the framework of special relativity with a universal speed of light c, the decay-length D can be converted to the decay lifetime x by

in any frame. In special relativity, both the decay-length and the lifetime a r e dilated. However, within the framework of taiji relativity with its use of natural units, one can only talk about the dilation of the decay length.

The dilation of

the decay length is completely relative within the framework of taiji relativity and special relativity. lOg.

Cosmic-ray muons and common relativity In common relativity, because all observers use the same set of clocks, t h e

time interval between the creation of a muon and its subsequent decay has the same value (measured in common-seconds) in all inertial frames.

Can common

relativity also predict correctly the result of the cosmic-ray muon experiment?

138

A Broader View of Relativity

Conceptually, it is important to understand the role of common time in this situation. Suppose that the F frame is at rest relative to Earth and that t h e r e is a muon n' at rest in the F' frame. In common time, the lifetime Atc'[u'] of this muon measured by an F'-observer is the same as the lifetime Atc[n'] of the same muon as measured by an F-observer on Earth At'chx'] = Atc[n'].

(10.30)

Now, suppose there is a second muon \x which is at rest relative to Earth and the F frame. In common time, the lifetime At'c[u] of this muon as measured by an F'-observer is the same as the lifetime Atc[u.] of the same muon as measured by an F-observer At'c[|l] = Atc[u].

(10.31)

Experimentally, we know that At c [u'] must be different from At c [u].

In

fact, those two time intervals must differ by a factor dependent on the speed of the muon in flight and this inequality implies that the time interval At'c[|x'] which represents the lifetime (in common seconds) of a muon at rest in F' measured by an F' observer, must differ from At c [n], which represents the lifetime (in common seconds) of a muon at rest in F measured by an F observer. Although this may appear at first to violate the principle of relativity, it actually does not because the principle of relativity implies only that the laws of physics be invariant in all inertial frames, and not the human convention used to define units. To clarify this point, consider the intervals Aw = A(btc) associated with t h e situation. Because this interval is associated with an actual physical quantity (i.e., the distance traveled by a light signal in a vacuum during a particular time interval) the principle of relativity implies that Aw'[u.'] = Aw[(i].

The

relationship between Aw[|x] and Atc[u] is determined by how human beings have defined the unit of time (second or common-second) relative to the unit of length. In common relativity, Aw[u.] and Atc[|x] are related by the factor c, Aw[u.] = cAtc[|x]. However, Aw'[u.'] and Atc'[u.'] are not so related because the speed of light is not isotropic in F'. Because the value of any intervals Atc are purely the

Experimental Tests II

139

result of how humans have decided to mark off the dials on a clock and to set its rate of ticking, the relationship between different Atc's is not governed by the principle of relativity. Instead, as we shall see in chapters 13-16, it can be set to whatever makes the calculations for the problem at hand the most convenient to perform. To see that common relativity makes the correct experimental prediction of the dilation of the decay lifetime of muons in motion, we must consider the interval Aw = A(btc), rather than Atc. The analysis then is identical to that under taiji relativity. If a muon n' at rest relative to the F' frame decays with a 'lifetime' Aw'[n'] as measured by an F' observer, then the decay 'lifetime' of that same muon Aw[n'] as measured by an observer in F is related to Aw'[(x'] by Aw[|x'] = Y AwV].

(10.32)

Since the principle of relativity implies that Aw'[[x'] = Aw[|i], i.e., the inertial frames F' and F are equivalent, then we obtain the familiar result from special relativity Aw[n'] =YAw[(i].

(10.33)

lOh. Quantum field theory and the decay length in common relativity Within the 4-dimensional symmetry framework of common relativity, the usual covariant formalism of perturbation theory can also be applied to quantum field theory by considering the interaction representation and the Smatrix. If one calculates scattering cross sections and decay rates (with respect to the lightime w=btc) of a physical process, one will get formally the same result as that in conventional quantum field theory. For example, one has the equations (10.20)-( 10.29), in which the taiji time w, if any, is replaced by the lightime bt c in a general inertial frame. To be specific, the mean decay length of an unstable particle in the decay process l->2+3+...+N is given by the decay rate r(l-»2+3+...+N)

140

A Broader View of Relativity

w-. -, o xn u m f kflSli>l2 d3x2d3p2 d3xNd3pN IX1-2+3+...+N) = w ^ J — ^ — -J2JJJ3- - ( 2 n J )3 >

w

=btc-

„„,„, (10.34)

The quantity r(l-»2+3+...+N) has the dimension of inverse length. Its inverse is particle's "lifetime" measured in terms of the lightime bt c which has the dimension of length and hence, may be called "decay length." The decay length D is given by D=l/r(l-2+3+...+N).

(10.35)

Thus, in common relativity, one has the "rest decay length" D0 for the decay of a particle at rest, corresponding to the "rest lifetime" multiplied by the speed of light in the conventional theory. The results in equations (10.34) and (10.35) imply that common relativity also predicts the dilation of the decay-length, consistent with experimental results.

Experimental Tests II

141

References 1.

H. E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938); 3 1 , 369 (1941). A hydrogen discharge tube is the source of the ions H2+ and H3 + .

After

acceleration, these ions could produce neutral but still excited hydrogen atoms, which emit the Balmer lines. 2.

The integer n is identical with the principal quantum number in n o n relativistic quantum mechanics. For detailed calculations for the h y d r o g e n atom, see J. J. Sakurai, Advanced Quantum Mechanics

(Addison-Wesley,

Reading, MA, 1967) pp. 122-127. 3.

The numerical value of p in taiji relativity (and common relativity) is t h e same as that of V/c in special relativity.

4.

We use the t symbol to avoid possible confusion with the ' symbol which is usually employed in the Lorentz transformations of two wave four-vectors such as those in (10.11). The symbols to and to' denote quantities of same atomic system as measured by different observers, i.e., F and F' observers. For a discussion of universal and fundamental constants in taiji and common relativities, see Appendix A.

5.

The dimensionless velocity fa in taiji relativity has the same numerical value as (vx / c ) for particle 1 in special relativity. See chapter 12 for a detailed discussion of the relativistic energy and momentum for a particle. Particle Data Group, Particle Physics Booklet (July, 1996, American Institute of Physics) p. 11.

A muon's mean life x is 2.197xl0" 6 seconds and its

decay-length is given by ct=658.654 meters. Recently, the lifetime (or the decay-length) dilatation predicted by the 4-dimensional symmetry of the Lorentz group has been tested and confirmed by measuring the decay lifetime of k° in flight at several hundreds of GeV (i.e.,y~10 3 .)

See N.

Grossman, K. Heller, C. James, et al, Phys. Rev. Lett. 59, 18 (1987). 6.

See for example, A. P. French, Special Relativity New York, 1968) pp. 98-103.

142

A Broader View of Relativity

(W. W. Norton & Company,

11. Group Properties of Taiji Relativity and Common Relativity lla.

General

group

properties

Lorentz and Poincare invariance is closely intertwined with the group properties of the 4-dimensional coordinate transformations. an

obvious

one-to-one

correspondence

between

Because there is

the

4-dimensional

transformations of taiji relativity (7.4) or common relativity (8.3) and the Lorentz transformation

(6.5), it seems mathematically

trivial

that

the

transformations (7.4) and (8.3) form a Lorentz group, just as the Lorentz transformations

do.

However,

transformations

(e.g., that t h e r e

the

new

concepts

are infinitely

behind

many

the

possibilities

taiji for

specifying the time t using a unit other than the meter through the relations w=bt and w'=b't') are highly nontrivial. In this chapter, we show that the taiji transformations, which are derived solely from the first postulate of special relativity, (i.e., the PoincareEinstein principle of relativity) have precisely the properties of the Lorentz group and the Poincare group. This is crucial 1 for the formulation of quantum field theories and particle physics on the basis of taiji relativity. Once the 4-dimensional group properties of the taiji transformations are established it follows that, regardless of one's definition of the ligh function b one always has a valid 4-dimensional symmetry framework with which to understand and investigate phenomena in the physical world. Mathematically, a group consists of a set of elements (or operators) G° = !go>gi>-"gk>--l

ar,

d an operation (also known as a "multiplication rule" and

denoted by a •) such that: (a) Combining any two elements gj and g^ using the operation leads to another elements g n in the set G°,

143

Si • gk = Sn-

(b) There exists a unit element g 0 in G° such that for any element gj, one has the relationship

go * Si = & * So = gi(c) For any element g^, there exists an inverse element gf = g"1 such that

g k ' g k 1 = g k 1 , 8 k = g0(d) The operation obeys the associative rule, ( g i , g k ) , g n = gi*(gk'gn)The most important groups in fundamental physics are those related to geometrical or physical transformations.2

For the Lorentz group, these

elements are the transformation matrices (11.15) below. The Lorentz group can be defined as the set of all 4x4 real matrices that leave the 4-dimensional interval w2-x2-y2_ z 2 invariant.

These matrices are singled-valued and

continuous functions of six parameters; three angles for rotations and three velocities for motion in 3-dimensional space. The values of these parameters are continuous, so that the Lorentz group is a continuous or Lie group. Let us consider the group properties of the taiji transformations w h i c h relate two inertial frames, F and F', which have mutually parallel axes.

First,

we consider the special case where the relative motion between the F and F frames is characterized by a dimensionless constant taiji-velocity p, measured in terms of taiji-time w, along parallel x and x' axes. The origins of F(w,x,y,z) and F'(w',x',y',z') coincide at the taiji-time w=w'=0. In addition to these two standard reference frames, let us also consider a third frame F"(w",x",y",z"), with axes similarly oriented, which moves with a constant taiji-velocity (pi,0,0) as measured in terms of the taiji-time w by observers in the F frame. simplicity, we shall call p or pi the velocity.

144

For

The constant velocity of F"

A Broader View of Relativity

measured by observers in F' is then denoted by p'^p^.O.O) and is related to p and pi by (7.8):

p'l = J _ ± 1-pp 1

p'^u'x,

px=ux.

(11.1)

In analogy with equation (7.4), the 4-dimensional transformations between F and F" are given by

w"= Yifw-pM,

x'^Yitx-pM,

y"=y,

z"=z;

(11.2)

1

l-(p¥ ' The inverse transformations of (7.4) can be easily obtained and are: w = Y(w'+px'),

x = Y(x'-pw'),

y = y',

z = z';

(11.3)

1 " P2

Based on (11.2), (11.3) and (7.8), one can show that the taiji transformations between F' and F" are w'^v'^w'-p^x'),

^-

p l

"

P

x'W^x'-p'V),

y" = y',

^-yyja-P1!.)-

1

z" = z';

(11.4)

'

Vi -(p' 1 ) 2

i-p ?

which have the same form as the taiji transformations between F and F', or F and F". Other

group

properties,

such

as

the

existence

of

an

identity

transformation [e.g., p=0 in (11.3)] , the existence of an inverse transformation

Group Properties of Taiji Relativity and Common Relativity

145

[e.g., (11.3) is the inverse transformation of (7.4)] and the fact that t h e transformations obey the associative rule, can be verified.

These properties

and the result (11.4) show that the set of 4-dimensional taiji transformations forms precisely the Lorentz group.3 These 4-dimensional group properties are critical to the ability of taiji relativity to predict correctly the results of all known experiments. This demonstrates that the Lorentz group can actually accommodate a wide class of different concepts of the time t (measured in a unit other t h a n meter) and that postulating the speed of light (in meters per second) to be a universal constant is not a requirement for transformations to form such a group. lib.

Lorentz

group

properties

In the previous section, we discussed group properties of the taiji transformations for inertial frames with relative motion solely along the x and x' axes.

In general, the taiji transformations relate quantities in inertial

frames with relative motion along x, y, and z directions that leave the 4-dimensional interval w 2 -x2-y2-z2 invariant.

Consequently, the contents of

the Lorentz group are much richer than what we have previously discussed. There are many different transformations which leave w 2 -x2-y2-z2 invariant. For example, there is a subset involving three discrete transformations: a spatial inversion (w -»w, r -» - r ) , a time inversion (w -» -w, r - » r ) and a spacetime inversion (w -» -w, r - » - r ) .

Another subset is a six-parameter

(three angles for rotations and three constant velocities for boosts) continuous group, which was first noted and discussed by Poincare in 1905. As stated before, the Lorentz group is defined as the set of all 4x4 real matrices that leave the interval s 2 invariant, where s2=w2-x2-y2-z2=T)|ivxlV=x|ix,i;

•n00=l.

x^=(w,r),

146

T

lll=1l22=Tl33=-1»

x =n„ v x v =(w,-r),

V

=

(11.5)

0

fOT

n,v = 0,l,2,3.

A Broader View of Relativity

M^V,

(11.6)

The metric tensors i ^ a n d

TI^V satisfy the relation TI„VTI^P = 6vP.

If x^ and xtj denote two coordinate 4-vectors (related by the taiji transformations), t h e n we have the following t h r e e types of

invariant

intervals in the 4-dimensional spacetime:

As 2 =(x 2 n-x 1 ) ( x / - x ^ ) > 0 :

a "time-like" interval,

(11.7)

As 2 =(x 2 -x1 )(x 2 M '-x/) = 0:

a "light-like" interval,

(11.8)

As 2 =(x 2 -Xj )(x 2 M '-x 1 ^)<0:

a "space-like" interval.

(11.9)

Physically, these invariant expressions describe the invariant laws of motion for three types of particles: (11.7) for massive particles, m 2 >0, (11.8) for massless particles, m 2 =0, (11.9) for tachyons (moving faster than the speed of light PL=1), m 2 <0. The first two laws have been experimentally demonstrated.

However, it is not

known why the third law (11.9) for "tachyons" is not physically realized in nature"^ or why tachyons, if they exist at all, have no interaction with ordinary particles. Let us consider the continuous Lorentz group with six parameters. With the help of tensor notation,^ the general 4-dimensional taiji transformations can be parameterized as follows, x,(i = A^ v x v ,

(11.10)

where we have employed the summation convention for repeated indices. Any quantity QV- with four components that satisfy the transformation (11.10) (i.e., QfeA^vQ;) is called a contravariant 4-vector. The transformation tensor A^v and the metric tensor r\ap satisfy the relation

Group Properties of Taiji Relativity and Common Relativity

147

•nnv^A^A^Tlap.

(11.11)

The metric tensors T}apand ri a Pcan be used to raise and lower the indices of any tensor. (For example, a covariant 4-vector Q^ is, by definition, related to the corresponding contravariant vector Q_v by Q_^ = ri^vQ_v and in general, one has A--HP....a-tlM.v^...vp....a ) The metric of the Lorentz group, given in (11.6), is ri|xv= (1,-1,-1,-1) which has three negative signs and one positive sign.

Thus, the

Lorentz group is denoted by 0(3,1). The 4-dimensional taiji transformation (11.10) can be written in matrix form and (A%) can be considered a 4x4 matrix.

(See (11.15) below.)

The

matrix form of (11.11) is ri=ATr|A, where the superscript T denotes the transpose of a matrix. The set of matrices {A}={A,A',A",...} leaves the interval s 2 in (11.5) invariant and can be considered elements of the Lorentz group.

Indeed, one

can verify that {A} satisfies the group properties mentioned previously in section 11a. The anti-symmetric operator LM-V of the Lorentz group was first discussed by Poincare in his Rendiconti paper. It is defined as L^=i(x^dv-xv^),

P-d/dXp.

(11.12)

It can be verified that L^v generates the algebra of the Lorentz group, [Ll xV )L ap ] = i(T1W3Lva

^a^if,

^ a ^

.^vPjJia^

(11.13)

We can parameterize the matrix A of the taiji transformations (11.10) in terms of (> by using equation (7.4): w' = Y( w - px) = w cosh (> - x sinh <> |, (11.14) x' = y(x - pw) = x cosh <>| - w sinh <(>,

148

A Broader View of Relativity

y = y.

•• cosh <> |,

Y =

vp = sinh <(>.

JM The Lorentz group is non-compact because the range of the parameter p does not include the endpoint 1. s 2 =g

If the metric tensor is g

=(1,1,1,1) [i.e.,

x^xv=w2+x2+y2+z2] instead of that in (11.5)), then the group would be the

4-dimensional orthogonal group 0(4) rather than 0(3,1). Since the coordinates (w,x,y,z) are real numbers, the group 0(4) is actually the rotational group.

4-dimensional

The parameters of the group 0(4) have finite ranges that

include the endpoints and hence the group 0(4) is compact.

Mathematically,

the 4-dimensional space associated with 0(4) is the perfect generalization of the ordinary 3-dimensional space with the distance x 2 +y 2 +z 2 . However, such a perfect generalization is not perfect from the physical viewpoint. The reason is that the invariant interval s 2 =g

x^x v =w 2 +x 2 +y 2 +z 2 has only a geometric

meaning and has nothing to do with laws of physics. This is in sharp contrast with the invariant interval (11.5) or (11.7)-(11.9), which is also applicable to the laws of physics. The taiji transformation (11.14) can also be written in matrix form

cosh(|> -sinh 0 0

-sinh<|> 0 0 cosh<|> 0 0 0 10 0 0 1

/ \

w X

w

= A(10,)

y

[z;

X

(11.15)

y z,

where the transformation matrix A(10,(t>) denotes a rotation in the wx (or xOx1) plane in the 4-dimensional spacetime. For an infinitesimal transformation, A, in (11.15) takes the form

A % = 5 % + E x!\,,

\»v

=-\v*.

(11.16)

The infinitesimal generator m 1 0 for the rotation in (11.15) is defined as

Group Properties of Taiji Relativity and Common Relativity

149

(o - l o o"! 10

m

-10 0 0 0

= - i —A(10,) I =-i i=o 0 d<|>

0 0 0 0 0 0

(11.17)

Similarly, for rotations in the x°x 2 and x°x 3 planes, the infinitesimal generators are given by

m

20

0 0 0 0 -10 0 0

-1 0 0 0

0^1 0 0 0

m

30

(0 0 0 - 1 0 0 0 0 0 0 0 0 -10 0 0

(11.18)

In equations (11.1)-(11.4), we see that two boosts in the x direction generate another boost. However, a boost in the x direction followed by a boost in the y direction does not generate another boost. rotation.

Instead, it generates a spatial

In fact, this is the physical origin of the Thomas precession.5

Therefore, we must also introduce three generators for spatial rotations in order to have a closed algebra.

This important property was recognized b y

Poincare, but not by Einstein, in their original work on relativity in 1905. The infinitesimal generators for spatial rotations are

,12

0 0 0 |0

0 0 0 0 10 - 1 0 0 0 0 0

,23

0 0 0 0 0 0 0 0 0 0 0-1

0) 0 1 0

(o o o o ,31

150

0 0 0-1 0 0 0 0 0 1 0 0

(11.19)

A Broader View of

Relativity

In general, the infinitesimal generator m^ v is defined to satisfy the relation mH^=-mvK An arbitrary infinitesimal taiji transformation can be written as V="°V

A(o,) = l + i v m ^

(1L20)

A finite rotation in the \nv plane (in the sense \n to v) is given by exponentiation A(nv, ) = exp(i<|>m ^ v ).

(11.21)

It can be shown that the generators m^v satisfy the commutation relation [m^,m a P] = i(ri^m v a + n v a m ^ -T^ a m v P -ri v Pm^ a ).

(11.22)

Let us define the spatial rotational generators J1 and boost-generators K* as follows: ji

=

l e ijk L jk ;

k

i=Loii

(1L23)

where L^v given in (11.12) are the generators of the Lie algebra of the Lorentz group. They satisfy the commutation relation (11.13) which is the same as that of m^v in (11.22). One can verify the following commutation relations: Ui,JJ]-leijkJk,

(11.24)

[K'.KJ] = -ie i J k J k ,

(11.25)

[ J S K J ] = ie^K*.

(11.26)

If one makes the following linear combinations of these rotational and boost generators,

Group Properties of Taiji Relativity and Common Relativity

151

Mk = - ( J k + i K k ) ,

(11.27)

Nk = I ( J k _ i K k ) ,

(11.28)

one can show that M and N commute, [M i ,N k ] = 0,

(11.29)

so that the algebra of the Lorentz group has been split into two pieces. piece generates a separate group called SU(2).

Each

One can use the irreducible

representations of SU(2) to construct representations of the Lorentz group.6 lie.

Poincard

group

properties

In order to have the most general transformations in flat 4-dimensional spacetime, one can generalize (11.10) to include translations such as x'^ = Aȣx v +b v ,

(11.30)

where b v is a constant and real 4-vector. transformations. (11.5) invariant.

These are known as the Poincare

Note that (11.30) does not leave the 4-dimensional interval Nevertheless, the interval

As2 in

(11.7)-(11.9) is still

invariant. The set of transformations (11.30) forms a Poincare group with 10 generators (3 angles for rotations, 3 constant velocities for boosts and 4 constants in b v for spacetime translations.)

Thus, in addition to the six

generators L^v given by (11.12), there are four translation generators P^ = i^^ (with a suitable choice of units, J=l). These generators satisfy the following commutation relations:

[L^v,LaP] = i(rf PL va + TivotL^P - r f a L v P - T ^ I T " ) ,

152

A Broader View of Relativity

(11.31)

VJ^KaVVV'

(1L32>

[P^,Pv] = 0.

(11.33)

This is the Lie algebra of the Poincare group (or the Poincare algebra). Based on these generators, one can construct two invariant (or Casimir) operators:

P„P^

and

W..W*,

(11.34)

where

w

n

=

i^vapp^

{1L35)

The tensor e^ v a P is the completely antisymmetric unit 4-tensor of the fourth rank and satisfies e 0 1 2 3 =l. Its components change sign upon interchange of any two indices, so that the components different from zero are equal to ± 1 . The two invariant operators in (11.34), P^P^ andW^W^, commute with all generators of the Poincare group.

The operator P^P^ has a clear physical

interpretation. Namely, it is the square of the mass of a particle. It is easier to see the meaning of \\y\VL\ if one makes a taiji transformation to the rest frame of the particle. In this case, the eigenvalues of P^ are given by (m,0,0,0) and we have

W°=0,

W^im

e^Ljk,

(11.36)

where WVm, i=l,2,3 are simply the usual rotational matrices in the three spatial dimensions which obey the angular momentum commutation relations. 1

the particle is at rest, W is just the spin operator.

Since

The eigenvalues of the

square of W=(wi) are therefore given by

Group Properties of Taiji Relativity and Common Relativity

153

W z = m z s ( s + 1),

(11.37)

m>0,

where s is the spin eigenvalue of a particle, s = 0, 1/2, 1, 3/2, 2,... . Thus, if m >0, a particle with spin s has 2s+l components (i.e., 2s+l independent states for a given momentum 4-vector). However, if m = 0, a particle with a spin s can have only 2 components.

For example, the photon has spin s=l, but only two

independent polarization states.

This property of the photon is a purely

kinematical property dictated by the 4-dimensional symmetry of Lorentz and Poincare groups. It is interesting that all particles in the physical world can be classified according to the eigenvalues of the two invariant operators in (11.34):

PP^O,

s = 0,1,2,3,... (bosons),

P„P^=0,

spin = ±s,

(11.39)

PP^O,

continuous s,

(11.40)

(tachyons).

(11.41)

PP* 1 < 0,

s = - , - , - , . . . . (fermions),

(11.38)

Particles with the properties in (11.40) and (11.41) have not been detected experimentally. Nevertheless, it is of great significance to the Poincare group that all particles (or fields) can be classified using the eigenvalues of the two invariant operators, one related to the mass of the particle and the other to its spin.7 In contrast, the Lorentz group does not have the translation operator P^ (or Vyi?^) and so is inadequate for the classification of elementary particles. Mass and spin are two fundamental properties of elementary particles. The fact that particles have a discrete spin can be understood on the basis of spacetime symmetry.^ Also, the CPT invariance for all interactions^ of particles is closely related to Lorentz invariance. However, the fact that elementary particles (such as quarks and

leptons)

also have discrete masses is far

understood. 10

154

A Broader View of Relativity

from

These discussions for Lorentz and Poincare groups hold for both the 4dimensional transformations in taiji relativity and common relativity.

Group Properties of Taiji Relativity and Common Relativity

155

References 1.

Although the form of transformations with 4-dimensional symmetry is simple, its significance to the formulation of particle physics

and

quantum field theory has not been fully appreciated by some authors who have attempted to modify the synchronization procedure for clocks a n d / o r to test a different ideas of time or the speed of light. For example, Winnie and Edwards attempted to obtain new transformations of space and time. However, the resultant transformations they obtained are untenable because the 4-dimensional symmetry of the Lorentz and Poincare groups was lost. See discussions in chapter 17. 2.

See, for example, P. Roman, Theory

of Elementary

Particles

(2nd. ed.,

North-Holland, Amsterdam, 1961), pp. 8-51; Wu-Ki Tung, Group Theory

in

Physics (World Scientific, Singapore, 1985) pp. 173-211. 3.

See, for example, J. D. Jackson, Classical Electrodynamics

(3rd ed., J o h n

Wiley & Sons, New York, 1999) pp. 539-548. 4.

For a discussion of tachyons, see G. Feinberg, Phys. Rev. 159, 1089 (1967); E. C. G. Sudarshan, in The Encyclopedia

of Physics (edited by R.M. Besancon,

Van Nostrand Reinhold, New York, 1974) p. 309 and references therein. 5.

J. D. Jackson, ref. 3, pp. 548-553.

6.

S. Weinberg, The Quantum

Theory of Fields, I. Foundations

(Cambridge

Univ. Press, New York, 1995) pp. 49-74; M. Kaku, Quantum Field Theory, A Modern Introduction 7.

(Oxford Univ. Press, New York, 1993) pp. 50-56.

The physical content of the Poincare group is richer than that of the Lorentz group and has not been fully explored.

See E. P. Wigner, Ann.

Math. 40, 149 (1939); E. Inonu and E. P. Wigner, Nuovo Cimento IX, 705 (1952); V. Bargmann, Ann. Math. 59, 1 (1954); S. Weinberg, ref. 5, pp. 55-81 8.

In the world of elementary particles and q u a n t u m fields, there is a necessary and fundamental connection between spin and statistics.

This

can be understood with the help of the 4-dimensional symmetry of the Lorentz group together with other requirements such as local fields, unique ground states, and microscopic causality. See J. D. Bjorken and S. D. Drell, Relativistic

Quantum

Fields (McGraw-Hill, New York, 1965) pp.

170-172.

156

A Broader View of Relativity

9.

For discussions of CPT invariance and weak interactions, see J. P. Hsu, Phys. Rev. D5, 981 (1972); J. P. Hsu and M. Hongh, Phys. Rev. D6, 256 (1972). 10. This suggests that symmetry principles alone are inadequate for understanding the phenomenon of mass and that perhaps, interactions of particles should be taken into consideration. However, it is puzzling that the electron and the muon appear to have exactly the same interactions (except for the "extremely" weak gravitational interaction), yet the muon mass is about two hundred times larger than the electron mass.

Group Properties ofTaiji Relativity and Common Relativity

157

12. Invariant Actions in Relativity Theories and Truly Fundamental Constants 12a.

Invariant

actions

for

classical

electrodynamics

in

relativity

theories Thus far, we have discussed different relativity theories such as taiji relativity, common relativity and special relativity. Later, in chapter 17, we will see how special relativity is a special case of extended relativity, which is based on the principle of relativity with a universal two-way speed of light. All these relativity theories possess the four-dimensional spacetime symmetry that is equivalent to the Lorentz and Poincare invariance. Therefore, we can develop Lorentz invariant actions for classical and quantum electrodynamics in these theories. In taiji relativity and common relativity, the speed of light (measured in units of meter per second) is either undefined or not an invariant quantity, so the invariant action for a free particle in, for example, the F' frame cannot

be

2

written in the usual form -/mc'ds', where ds' =dx'^dx'^. Nevertheless, mds'=mds is invariant because the interval ds (=ds') and the (rest) mass m are scalars. In all relativity theories, regardless of whether or not the speed of light (in meter p e r second) is universal constant, the invariant action SCED for classical electrodynamics (CED) can be assumed to take the following form, 1

SCED= J l - m d s - e a ^ d x ^ - i J > % v d 3 r d w ,

xM- = (w,r),

x,x =nnVxv = (w,-r),

ds 2 = dx(idx(i = [dw] 2 - [dr] 2 ,

(12.1)

^=3^-**^,

e = -1.6021891xlCr 20 -\/4Jt

158

(gcm)1/2,

where x^ is the coordinate 4-vector, av is the electromagnetic 4-potential, and ds2 is the differential form of the 4-dimensional interval. The electric charge e is a universal constant, which happens to be the same thing as the charge expressed in Heaviside-Lorentz (rationalized) units, If the time variable t (measured in seconds) is introduced in a relativity theory, then one has w=bt and w'=b't', where b is the ligh function (e.g., in special relativity, b=b'=c). The action (12.1) has different dimensions from the usual action in special relativity. Comparing (12.1) with the corresponding action S sr in special relativity for the F frame, we see that S=S S r/c,e=e/c (e in esu) and ali=atl(w,r)=A(1(w,r)/c, where A^ is the usual electromagnetic vector potential in special relativity. Note however that (12.1) is more general in that it can be used as the action in any relativity theory, including special relativity. In preparation for the discussions in chapters 13-16 related to common relativity, let us first consider the action (12.1) within the framework of common relativity. As mentioned before, common relativity has two universal constants in classical and quantum electrodynamics, J and e, given in (10.7). In a general reference frame, the 4-coordinates are denoted by (w,x,y,z), where w=btc. The invariant action for a charged particle interacting with the electromagnetic field is given by the first two terms in (12.1):2 SC = J {-mds - e a ^ } = j Lcdtc ,

Lc = -mVC2-v2- ea0C + ea-v,

a^ = (a0, -a),

(12.2)

^ - = Vc 2 -v 2 , C = ^ ^ - .

In the following discussion, we use C, which is in general not a constant, as a shorthand for d(bt c )/dt c . The canonical momentum P and the Hamiltonian Hc for a charged particle are defined, as usual, by _

dLc

P = —-= p + ea, 5V

(12.3) H c = [(3Lc/av)-v - Lc]=CpQ+ eaQC,

Invariant Actions in Relativity Theories

159

m

mv/C

3„= • — 0

, 2

Vl-V /C2

p =

^

„

?

P^Pu = m z ,

,

P P

V1-V2/C2

*

where p^=(p°,p) and p (i =(p 0 ,p 1 ,p 2 ,P 3 )=(p 0 ,-p) are the 4-momenta of the particle. Note that the canonical momentum (P°,P), where P°=HC/C, also forms a 4-vector. The Lagrange equation of motion for a charged particle in the electromagnetic field can be derived from (12.2) by the variational calculus. One has

£-ef^. P

dx^ ~mds

/

(12.4)

m mv/C \ '^y/l-v2/c2'yll-V2/C2^

where p(i=(p°,p) is the usual momentum for a particle. Multiplying both sides of (12.4) by ds/dt c , one obtains the equation of motion expressed in terms of the common time t c ,

dP^ef^^, *c dt,

(12.5)

c

^

do 0 ^H

E

-

V

>

do a^e(EC

+

vxB),

where p° and p are given in (12.4) ande(EC+vxB) is the Lorentz force in common relativity. For a continuous charge distribution in space, the second term in (12.1) becomes - J a (l J ,i d 3 rdw , where w=bt c .

The invariant Maxwell equations in a

general inertial frame in common relativity are thus

160

A Broader View of Relativity

x x = (bt c ,r),

f^v= d W - dva.v-.

dx* One can write the field-strength tensor fM* in matrix form / 0 Ex Ey \ Ez

f l*v

-E x -E y -E z \ 0 - B z By Bz 0 - B x -By B x 0 /

da a(btc)

V a0,

(12.7)

B = V x a.

(12.8)

Since a^(w,r)»A^(ct,r)/c, the fields E(w,r) and B(w,r) are related to the usual EsR(ct,r) and BsR(ct.r) in special relativity by the following correspondences: E(w,r) •» EsR(ct,r)/c,

B(w,r) <* BsR(ct,r)/c.

(12.9)

The transformations of the field tensor f^v are given by

f |iv _

aA

g

* fa|3 3X dxP

(12.10)

a

where dx'^/dx a can be calculated from the coordinate transformation (8.3). Since the fields E and B are elements of the field tensor f a P, the transformation equations for the electromagnetic fields are E' X =E X ,

E' y = v(E y -pB z ),

E'z = Y(Ez + PBy),

B' X =B X ,

B' y = Y(By + pEz),

B' z = Y (B z -pE y ).

(12.11)

Note that one can use transformations (8.13) and (12.11) for velocities and electromagnetic fields to show that

eE-v and the Lorentz force

Invariant Actions in Relativity Theories

e(EC+vxB) in

161

(12.5) form a 4-vector (eE-v, e[EC+vxB]), i.e., they transform like a 4-vector. In terms of E, B, and the 4-current J^=(p,J), the first equation in (12.6) can be written as

V-E = p,

V x B - -J2—-J. a(bt c )

(12.12)

The second equation in (12.6) can be written as

V-B = 0,

dB

VxE +

=0.

(12.13)

a(bt c ) In common relativity, the speed of light does not explicitly appear in the new formulation of the Maxwell equations in a general inertial frame. The Maxwell equations can be written in the familiar form

V-B = 0,

VxE-

-—= c at

0,

etc.,

(12.14)

if and only if the speed of light is assumed to be a constant c.

Nevertheless,

when p = 0 and J = 0, the wave equations (12.12) and (12.13) can be written in the 4-dimensional form, 2 i_£._V2g = 0, flw2

w = btc,

(12.15)

where the function g=g(bt c ,r) stands for any component of E or B. A plane wave solution g p of (12.15) takes the form

gp= A o s i n ( k 0 w - k r ) ,

kg-k2=0.

(12.16)

When y=z=0, the wave equation (12.13) with g=g(bt c ,x) leads to the general solution in any inertial frame, g(bt c ,x) = gi(x+bt c ) + g2(x-bt c ), where gi and g2 are two arbitrary functions. Since b is in general, a function, it is not the speed

162

A Broader View of Relativity

of the wave. To obtain the speed of the wave, we may consider the amplitude g2 which satisfies the equation x - bt c = constant. Differentiating with respect to the common time t c , we find the speed of this wave to be dx

d(bt c )

$-c=-i*r=c>

(12 17)

-

where the speed of light C as measured by the common time t c in a general inertial frame is not isotropic, as shown in equation (8.5). Only in the inertial frame F(ctc,x,y,z) does one have a constant speed of light b=C=c. 12b.

Universal

constants

and

invariant

actions

In taiji relativity, where we use natural units, the only constants that appear in the equations are dimensionless constants such as a e .

All other

constants such as c, ti, etc. turn out to have the value 1 (see Appendix A for a more detailed discussion). In common relativity however, in which we define the common-second as our unit of time tc, universal constants with units exist. In this section, we examine what they are and compare them with the usual ones found in special relativity. According to quantum mechanics, the wave 4-vector k^ of a quantum particle must be proportional to the 4-momentum p^. Thus, in general we have

pV=j'k'n,

in F'frame;

(12.18)

p^ = J k i \

in F frame.

(12.19)

Based on symmetry considerations, the same proportionality relation should hold in any inertial frame so that J'=J. Therefore, the constant J is a universal constant in common relativity. 1 Its value can be obtained by comparing (12.19) in F with the usual relation in special relativity

(SR) p^sR=^k^,

where

P^SR/C<S>P^, k°SR=eo/c«>k0 and kSR=k in F. Note that the last relation must be true

Invariant Actions in Relativity Theories

163

because the spatial components r in common relativity are the same as the usual rSR. As a result, we can deduce that the value of J must be

h/c=h/(2nz):^

J = 3.5177293 x 1 0 " 3 8 g e m .

(12.20)

This result is consistent with (10.7) and (10.15) in chapter 10. The quantity J is unusual because it is a universal what one uses for a second postulate in the 4-dimensional Equivalently, it is independent meter) for the evolution at all, it is a constant. constants

dictates

variable.

i.e., the one chosen

by Einstein

between

on a particular

than

definition

to the conventional

depends

of

framework.

unit (other

As long as one makes any kind of

This is in sharp contrast

a certain relationship

regardless

symmetry

of how one defines an additional

c and h, whose universality

second postulate,

constant

universal choice for a

for special relativity,

which

w and t. The universal constant J in

common relativity plays the role of the Planck constant in the conventional theory.

For example, we have expressions such as p = - i j v , exp(ipM,x^l/J),

d 3 rd 3 p/(2jij) 3 and [J 2 (a 2 /dw 2 )-J 2 V 2 -m2]
equation

(12.3) with the

relativistic relation in the F frame, one has

corresponding

special

e=e/c (where e is measured in

electrostatic units, esu,) and a^(w,r) <* A^(ct,r)/c; the latter is necessary, so that the action for the free electromagnetic field -(l/^Jf^f^rdw

,

fnV = a l i a v -a v a ( „

(12.21)

has the same dimension as Sc in (12.2), as is required for consistency.

Thus, the

universal constant e in common relativity is e =-1.6021891x10-2° (4JT) 1 / 2 ( g c m ) 1 / 2 .

(12.22)

Like J, e=e/c is a universal constant in all relativity theories, regardless of the choice of second postulate.

It can be verified

t h a t the

dimensionless

electromagnetic coupling strength a e has the usual dimensionless value of a e = e 2 /(4jtj) = 1/137.0359895.

164

A Broader View of Relativity

Note that although from the conventional point of view of special relativity, there are three universal constants h, e and c in quantum electrodynamics, we see that in relativity theories in general, there are only two such constants, J and e, the universality of which are independent of the specific transformation property of the evolution variable. In other words, they are independent of how the unit of time t is defined. 12c. Dirac's conjecture regarding the fundamental

constants

In his 1963 article "The Evolution of the Physicist's Picture of Nature", Dirac gave an interesting account of how physical theory had developed in the past and how it could be expected to develop in the future. Dirac noted that in the conventional framework of physics based on special relativity and quantum mechanics, three fundamental constants in nature are the charge e, the Planck constant h, and the speed of light c and that using these constants, one can construct a dimensionless number ae=e2/(4nhc), which has a value very close to 1/137. Dirac wrote that "there will be a physics in the future that works when hc/e2 has the value 137 and that will not work when it has any other value." (The factor of 4n in a e is not important since one can choose suitable units such as Gaussian units for the charge e such that a e does not contain the factor of An.) Dirac then gave the following outlook on physics: "The physics of the future, of course, cannot have the three quantities h, e, and c all as fundamental quantities. Only two of them can be fundamental, and the third must be derived from those two. It is almost certain that c will be one of the two fundamental ones. The velocity of light, c, is so important in the four-dimensional picture, and it plays such a fundamental role in the special theory of relativity, correlating our units of space and time, that it has to be fundamental. Then we are faced with the fact that of the two quantities h and e, one will be fundamental and one will be derived. If h is fundamental, e will have to be explained in some way in terms of the square root of h, and it seems most unlikely that any fundamental theory can give e in terms of a square root, since square roots do not occur in basic equations. It is much more likely that e will be the fundamental quantity and that h will be explained in terms of e2.

Invariant Actions in Relativity Theories

165

Then there will be no square root in the basic equations. I think one is on safe ground if one makes the guess that in the physical picture we shall have at some future stage e and c will be fundamental quantities and h will be derived." 3 We can compare Dirac's outlook in 1963 with our present understanding of relativity and of the 4-dimensional symmetry of physical laws. Because one can formulate an experimentally consistent theory based solely on the first postulate of relativity (using the same unit to quantify both space and time), and because the constant c (with units of meter/second) does not appear in such a theory, one sees that the constant, c=299792458 m/s, is not necessary to the formulation and understanding of physics. Dirac's conjecture implicit assumption

that the evolution

(or temporal)

was based on the

variable must be

with a unit different from that used for spatial variables.

measured

As our understanding

of physics has progressed, we have seen that this turns out not to be necessary. It is precisely because c merely correlates our units of space and time that its value is not a universal and fundamental constant. As Taylor and Wheeler state in Spacetime committees."

Physics, the value of c is simply "the work of two centuries of 4

More fundamentally, the quantity that correlates the evolution

and spatial variables is simply 1, which is the dimensionless and universal taiji-speed of light derived from Maxwell's equations in taiji relativity. 12d.

Truly fundamental

constants

At this point, one might ask: If the speed of light c is neither a necessary nor a truly universal constant, why do all experiments (before 1983) that measure the speed of light in different inertial frames give the same value c=299792458 m/s?

To answer this question, let us analyze the origin of the value

of c from the viewpoint of the Poincare-Einstein principle alone, i.e., in the framework of taiji relativity. Suppose that a physicist introduces the time t' as the evolution variable in an F' frame. This time t' must have some relation to w', say w'=b't', as required by 4-dimensional symmetry.

In taiji relativity, the

invariant phase of an electromagnetic wave can be written as

k^x'n = 4 b ' f - k ' T ' , where C'=d(b't')/dt',

166

A Broader View of Relativity

(12.23)

where C and b ' are functions which are separately undefined.

These two

functions are different but related, because the speed of light C is given by the invariant law of propagation (7.9) in the frame F' with w'=b't', [d(b't')] 2 -dr' 2 s (C'dt') 2 -r' 2 =0, i.e., d(b't')=C'dt'. Now, suppose our physicist chooses to measure time t' using the electromagnetic frequency co'. At first glance one may t h i n k that this can be done without making any assumptions about the speed of light, but this is not true. In order to measure the time t' using the frequency co', one must adopt a specific relationship between the two quantities.

Typically, the

relationship that is adopted is the one that assumes that the invariant phase has the more restricted form, to't'-k'-r' and thus that the electromagnetic oscillation has the form sin(co't'), as it does in special relativity. This amounts to making an additional assumption that C'=b'=c=constant in the i n v a r i a n t electromagnetic waves.

phase

In taiji relativity, co' and t' have no

of

specified

relationship (indeed, t' does not even exist), and thus one cannot be used to measure the other. With the pre-1983 definitions of meter and second, the speed of light that results experimentally from the aforementioned hidden assumption turns out to have the value 299792458 m/s. Therefore, the fact that previous experiments seems to imply a particular universal value is merely the result of a subtle assumption that fixes a relationship where no a priori

relationship

existed, combined with the historical definitions of our units of length and time. If one looks at the physical world from the simplest viewpoint, i.e., based solely on the 4-dimensional symmetry of the Poincare-Einstein principle alone, electromagnetic waves have two types of periodic behavior, one related to our perception of spatial length determined by Ikl (or the wave length X=2jt/lkl) and the other related to our perception of "duration" determined by ko, as shown in (12.16). Since lkl=ko, then it is clear that both spatial and temporal intervals can be measured using the same unit and these waves propagate with a dimensionless speed of 1. In other words, the relationship between units for the spatial interval Ax=l m and the "time" interval Aw=l m are completely dictated by the Maxwell equations and the Poincare-Einstein principle.

Since our

ancestors were not aware of the Poincare-Einstein principle and their intuitive sense of time was different from that of space, they invented a quantity with a different unit to measure time intervals. As a result, we have a quantity called

Invariant Actions in Relativity Theories

167

frequency GO measured in the unit of 1/second and it is purely a matter of history and human convention that the duration At=l s is equivalent to the taiji-time interval Aw=299792458 meter. If the numerical value 299792458 m/s is not an inherent property of our universe it is not a fundamental physical constant. Thus, we conclude that there is only one truly universal and fundamental constants in classical electrodynamics, the dimensionless fine structure constant a e = 1/137.0359895. Based on a similar line of reasoning, we further conclude (see chapter 23 and Appendices A and C) that a e is the only truly fundamental physical constant in quantum electrodynamics in both inertial and accelerated frames, in contrast to Dirac's conjecture.

168

A Broader View of Relativity

References 1. 2.

3. 4. 5.

Jong-Ping Hsu and Leonardo Hsu, Phys. Letters A 196, 1 (1994). Jong-Ping Hsu, Phys. Lett. A 97, 137 (1983); Nuovo Cimento B 74, 67 (1983); J. P. Hsu and C. Whan, Phys. Rev. A 38, 2248 (1988), Appendix. These papers discuss a four-dimensional symmetry framework based on the usual first postulate and a different second postulate, namely, a common time t'=t for all observers in different inertial frames. P. A. M. Dirac, Sci. Am. 208, 48 (1963). E. F. Taylor and J. A. Wheeler, Spacetime Physics (Second Ed., W. H. Freeman and Company, 1992), p. 58. For detailed discussions, see chapter 23, and Appendices A and C.

Invariant Actions in Relativity Theories

169

13. Common Relativity and Many-body Systems

13a. Advantages of common time In classical mechanics, the motion of a many-body system can be described in terms of the position of the center of mass r c , which has a well-defined properties under the Galilean transformation. However, in the mechanics under special relativity, the concept of the center of mass for a system of particles is not well-defined because the moving mass is not a constant. The center of energy RCE defined by

i c tot

k

in special relativity corresponds to the center of mass r c in classical physics, where E; and r; are the relativistic energy and position of the ith particle, respectively. Clearly, RCE reduces to r c in the nonrelativistic limit because Ej becomes the mass m;. However, the components of RCE do not transform as components of a 4-vector under the Lorentz transformations.

A similar

situation occurs with the total momentum P,i=2Pi,x. i

which, in general, is not a 4-vector. The reason is that the sums P^ are, in fact, sums of the values of p;^ at a given instant in the given frame. However, in a different frame these values are no longer the momenta of each of the individual particles at the same instant. Therefore, when one transforms the total momentum P^ to a new frame F', one has not only to form

170

2PV > i

according to the 4-vector rule, but also must recalculate these quantities appropriately according to the simultaneity of the F' frame. 1

Only for

noninteracting particles is the sum P^ a 4-vector because each pi^ is constant. Apart from these problems, the task of characterizing many-body systems has other difficulties within the framework of special relativity: 2 (i) The concept of a Hamiltonian system with many degrees of freedom breaks down because of the lack of a single time for a system of N particles. Relativistic time appears to be incompatible with the notion of the canonical evolution of a many-body system and with the evolution of a stochastic process. (ii) Since the notion of a 3-dimensional spatial volume is not a covariant concept, the uniform density in configuration space is not normalizable. These properties lead to several difficulties in treating

many-body

systems within the framework of special relativity because, in contrast to Newtonian mechanics, it does not have a single time which is applicable to all particles or to all inertial frames of references. 2 ' 3 Common relativity, on the other hand, does not have these difficulties. is the natural three-dimensional

and unique framework

four-dimensional

generalization

of the

It

classical

with one single time. As demonstrated in chapter

12, common time is universal but differs from the Newtonian absolute time in that it also is consistent with the principle of relativity.

It provides a universal

simultaneity, just like the absolute time in classical mechanics.

As a result,

common time embedded in a 4-dimensional symmetry framework enables us to formulate the Hamiltonian dynamics of a many-body system and also to define an effective and invariant 6-dimensional (x-space and 6N-dimensional phase space for a covariant statistical mechanics, which special relativity does not allow. 3 The assumption of common time in space-lightime necessitates

the

existence of the ligh function b in a general inertial frame, b = xo/t c = w/t c , which transforms as the zeroth component of a 4-vector because the common time t c is a scalar. This is important because the ligh function b enables us to define an invariant volume Vi in a general inertial frame:

Common Relativity and Many-Body Systems

171

V I = f — d 4 x5(t c -— )= f — dxdydz, J c0 b J c0

coSc,

(13.1)

where d4x=dx0dx1dx2dx3=dx°dxdydz and the scaling constant c 0 sc is introduced to preserve the usual dimension of the volume Vi. Note that there is no restriction on the zeroth component x°in d4x=dx°dxdydz, just as x°=w in equations (7.1) and (7.4) within taiji relativity. The 6-function in (13.1) imposes on x° the restriction that one uses common time in all inertial frames. Note that the condition imposed by the 5-function in (13.1), i.e., t c -x°/b=0, is invariant. This invariance can be seen as follows:

. _ x°

_x*M_ft b'N

M-fl^l [

N-(l + J*

X'0

(13.2)

From the expression for N and the constraint [tc - x'°M/(b'N)] = 0, one can obtain the relation between M and N, N=l/[l-px'/(x'°M)]. Thus, one has t

X c

' ° M _ t _ x'°M[l-px'/(x,0M)] c b'N b' _. = tc

[x'°(l-px'/x' 0 )-px')] _

x'° tc- — .

(13.3)

It follows from (13.2) and (13.3), that the volume element dVi= c0-1d4x5(tc - x°/b) =(lbl/c0)dxdydz is invariant under the 4-dimensional coordinate transformation in common relativity. The numerical value of c 0 in a general inertial frame is defined to be identical to the speed of light in F, c0=c=b, so that Vi is the same as the usual volume in the frame F. Even though the speed of light is not isotropic in a general inertial frame F', one can still make this definition without violating the 4-dimensional symmetry of common relativity because c0 is simply a scaling constant. In the F'-frame, an invariant integral involving dVi=(lb'l/c)dx'dy'dz' is, in general, not convenient for carrying out calculations because of the presence of the ligh function b'. However, we can always transform quantities to the F-frame, in which b=c and dVi=dxdydz, for ease of

172

A Broader View of Relativity

calculating invariant integrals. One can also understand the new invariant volume dVi in (13.3) as the volume of the parallelepiped in the space-lightime defined by four vectors a^, b v , c a , and dp: dVj = E ^ a P a ^ b v c a d p )

(13.4)

where a,Hb/co,0,0,0), b v =(0,dx,0,0), c a =(0,0,dy,0) and dp=(0,0,0,dz). It is the same as the invariant expression (13.4) provided a^, b v , c a , dp are vectors lying in the direction of the coordinate axes and of length dw, dx, dy, dz, respectively.

The

hypersurface dzv in the 4-dimensional space-lightime, d2^=(ds 0 ,d2 1 ,ds 2 ,ds 3 ),

(13.5)

transforms like a 4-vector, where dz°=dxdydz is equal to the three-dimensional volume element. Geometrically, the 4-vector dff is equal in absolute magnitude to the "area" of the element of hypersurface and points in the direction perpendicular to all lines lying in this element. The existence of the invariant volume Vi is important to four-dimensional thermodynamics and statistical mechanics. It enables us to define the notion of a box (or a 3-dimensional spatial volume) and to normalize the uniform density in configuration space. 13b.

Hamiltonian

dynamics

in

common

relativity

In order to see what new quantities are covariant under the common relativity transformation and properties of the invariant 6N-dimensional phase space, let us briefly consider kinematics and the invariant

Hamiltonian

dynamics. The invariant "action function" for a free particle is assumed to be S = -J*mds = -J"m-Jv l % dt c = J*Ldtc, L = -m^|v^v^ = - m V c 2 - v 2 , where ds 2 =dw 2 -dr 2 =(C 2 -v 2 )dt 2

and C=dw/dt c =d(bt c )/dt c in a general

Common Relativity and Many-Body Systems

(13.6) inertial

173

frame. The 4-momentum p^ of a free particle is defined as • -^-= , m v " / C av

n

2

Vl-v /C

2

= Gv>J,

Vu=(C,-v), vi* = (C,v),

(13.7)

where p^ has the dimension of mass. This definition is consistent with the usual definition of the momentum p=aL/av. Although mvi* is a 4-vector, it is not the physical momentum. The quantity G,

G- ,

m =

Vc 2 -v 2

T/P!±JH1=P^ C

(13.8)

C

is an invariant, being the ratio of corresponding components of two 4-vectors, p^and vM-, with p x /v x ^> 0 /C. Its invariance can also be seen from (13.7) because pi* and vv- are 4-vectors with the same transformation properties in common relativity. We call G the "genergy" because it is equivalent (except for a multiplicative constant) to the conserved energy p° in the F-frame, in which C=c so that the conservation of energy implies the conservation of the invariant genergy. Since special relativity does not have a four-velocity measured in terms of a scalar common time, it does not have an invariant quantity analogous to the genergy (13.8). With the help of the invariant genergy G, we can define the four-coordinate R£ of the "center of mass" for an N-particle system in common relativity, 0,1** R^=^—, Z^k k

N

Q.1" = ^ G p f .

(13.9)

i=l

The spatial components Rc can be satisfactorily identified with the relativistic center of mass for a system of particles in common relativity. It can be verified that Rc reduces to the classical center of mass when the velocities of the particles are small. Similarly, the total momentum p

, = 2>in

174

A Broader View of Relativity

in common relativity is a 4-vector because the common time gives us a universal definition of simultaneity which is applicable to all observers in all inertial frames of reference. Let us consider N charged particles with masses m a , a=l,2,..., N, within the framework of the common-relativistic Hamiltonian dynamics.3 We begin with an 8N-dimensional "extended phase space" with the basic variables x^,p*, to define the Poisson brackets and the generators of the Poincare group. {xg.x^J-O,

{xg.p^-TTSab,

IpS.pp-a

(13.10)

where {A,B}*

, flx£

aPan

W^

x^=(wa,xa,ya,za).

ax ^

We impose 2N constraints of the form Ka-(P£-eag)2-ml=Q

Z a (xg,t c ) = x 2 - b a t c =0,

where P^ is the canonical momentum 4-vector, P a = P a - § a a >

(13.11) an

d a£ is the

electromagnetic potential acting on the particle a. Thus, we end up with 6N basic independent variables. The Dirac Hamiltonian is H D =W a K a ,

(13.12)

where W a is determined by the constraints Za=0 which must hold for all times t c , i.e. dZ/dt c =0. The Hamiltonian equations of motion with the common time t c are then 3

M dt c

= {x lt,H D }

= iMD. i apan

M dt c

=

|p£,H D } = - ^ , dxa[l

(13.13)

where Wa=-Aabft,

A ab !Z b ,K c } = 8 a c .

ot r

Common Relativity and Many-Body Systems

175

One can verify that Hamilton's equations of motion (13.13) with the Dirac Hamiltonian HD in (13.12) and the constraints (13.11) lead to the equations of motion for a charged particle,

— = e fa^Vav. dtc v

av

a = l,2,...N,

(13.14)

_ Ca(Pav ~ ea av) j; » .

(P°-ea°) dZa(x°, tc) = dx° - ^ a l d 6

A

"

dtc

dtc

- dx° - Cadtc,

^ i = _ca> dt c

ac

2(P°-ea°)'

2(P a °-ea°)' where a is not summed. The resultant equation of motion (13.14) is the same as that in (12.5), as it should be. Thus, the evolution of the N-particle system in terms of the common time t c is completely determined by Hamilton's equations (13.13) and the initial data, i.e., the initial momenta and positions of the particles. A basic unsolved problem in the statistical mechanics based on special relativity is related to the phase space in different frames of reference. In a general inertial frame, it appears to be impossible to have a meaningful Lorentz-invariant phase space related to the initial data at a given time for all observers in different frames, in contrast with the classical case based on Newtonian laws. In Newtonian physics, observers can, in principle, perform measurements at a given absolute time. Thus one can specify a system of N particles by the vectors (qi,...,qN.Pii—>PN) which span the phase space of the system. In common relativity, suppose that there is a clock and an observer at each point in space in every inertial frame. The F-frame observers can perform instantaneous measurements of r and v for each particle at a given

176

A Broader View of Relativity

common time t c using the nearest clock so as to remove the need to account for the finiteness of the speed of light. Thus, we have the coordinate and momentum data of particles (r a ,p a ), a=l,2,..., N, at the common time t c . These data, along with the dynamical equations, completely determine the evolution of the system. Of course, the data can also be transformed to another frame F' using the space-lightime transformations which relate (ctc, r a )

and

(c,va)

in F,

a=l,2,..., N,

in F',

a=l,2,..., N.

to (b'atc, r' a )

and

(c' a ,v' a )

The initial positions and velocities of the particles in F can also be directly obtained by the F' observers by measuring r ' a , v ' a and computing b' a and c' a =d(b' a t c ))/dt c with the help of eq. (8.3). In this way, we have global knowledge of the whole system at a given instant of common time in the statistical mechanics based on common relativity. With the help of (13.3) and (13.11), we can define in (i-space an effective six-dimensional volume element (for particles with maO) dn = d 4 x 5(-—t c )d 4 p S(p£-m 2 ) 9(p°)2G. b

(13.15)

This volume element is an invariant because the common time tc, the rest mass m, the genergy G of a particle, and the ratio Xo/b are all scalars in common relativity. In the particular frame F, one can see that eq. (13.15) reduces to d3xd3p by carrying out the integrations over x° and p°. The 6N-dimensional phase space in common relativity has the invariant volume element N

dr=f]d[A a .

(13.16)

a-l

The independent physical variables in phase space are the only really important ones. We can eliminate the nonphysical variables from the theory using the constraint equations. However, this elimination may be awkward and may spoil the four-dimensional properties of the equations; therefore, we shall retain the constraint equations in the general theory. That is, we will treat the

Common Relativity and Many-Body Systems

111

N-particle phase space as an invariant 8N-dimensional extended phase space with 2N invariant constraints (13.11).

In this way we have, effectively,

a

physical 6N-dimensional phase space that is invariant under the space-lightime transformations. 13c.

Invariant

kinetic

theory

of

gases

Let us consider kinetic theory based on a one-particle distribution function D^x^tc) and a kinetic equation for D!(x A ,t c ) within the framework of common relativity. In order to show explicitly the 4-dimensional symmetry of the theory of gases we use xA=(x^,p^), where p H s the 4-momentum of the particle in (13.7), as the coordinates of the extended n-space, i.e., the one-particle phase space. The reason for this is that the 4-dimensional transformations of coordinates can only be expressed in terms of the Cartesian coordinates xK The invariant time-dependent distribution function D^x^tc) in the n-space is normalized by the expression /Di(x A ,t c )dn = /D*d 4 xd 4 p = N , (13.17) 2

D* s Di(xA t) 8(t c - — ) b(px - m2) 6(p°)2G , b where duTs given by (13.15) and N(t c ) is the number of particles in the gas at time t c . The function D*d 4 xd 4 p/N can be formally interpreted as the probability of finding the particle at common time t c in the volume element d 4 xd 4 p around the point x A in the 8-dimensional space. If one carries out the integration over x° and p°

from -<*>to +°°, then [(l/N)jD*dx 0 dp°]d 3 xd 3 p is the probability of

finding the particle at time t c in the spatial volume element d 3 x around the point r and with a momentum in the region d 3 p around the value p. This is in sharp contrast with the usual relativistic kinetic theory in which the distribution D' is normalized on a six-dimensional manifold t h r o u g h a current 4 /D'2 P(i e(p°)5(p£ -m 2 )d 4 pdz,i - N,

178

A Broader View of Relativity

(13.18)

because of the lack of a single evolution variable for a system of particles.

In

other words, the usual relativistic kinetic theory is a statistics of curves rather than a statistics of points. In the conventional theory, such a six-dimensional manifold is, in general, not meaningful because of the arbitrariness of 2 (except the case when the current is conserved) and the fact that the distribution D' is not actually the probability density. 2 The invariant distribution D!(x A ,t c ) is a function in the eight-dimensional space with properties

as defined

by (13.17).

The

common-relativistic

one-particle Liouville equation for the invariant distribution D 1 (x A ,t c ) can be derived by considering the eight-vectors x A (t c ) of the particle coordinates, the corresponding velocities x A (t c ) =

dx A (t c ) dtc

= (xH(tc), pH(tc)) = (v(\ FC) ,

(13.19)

and the current density j A (x A ,t c ), j A (x A ,t c ) = D 1 (x A ,t c ) x A (t c ).

(13.20)

The current density j A (x A ,t c ) satisfies the continuity equation in the eightdimensional space ^ D 1 ( x A , t c ) + -L-[D 1 (x A ,t c )x A (t c )] = 0 . 3t c axA

(13.21)

Using Hamilton's equations of motion (13.13), we have _£_x A (t c ) = _ ^ « H D axA dx* a P|.

+

_J_. ^ B l = 0, Pn I dx»

(13.22)

a

and therefore, we can write (13.21) as —D!(x A ,t c )+ [ v ( * — + F H — ] D 1 ( x A , t c ) = — D 1 (x A ,t c )=0. at c

L

dxii

J

gp\i

(13.23)

dtc

This is a simple example of the Liouville theorem, to be discussed in section 13d

Common Relativity and Many-Body Systems

179

below. Clearly, if the one-particle distribution does not depend on t c explicitly, i.e., Dx = D!(xA), then we have an invariant distribution Dx(xA) which obeys equation (13.23) where the first term is zero. In order to see the connection to the nonrelativistic case, we write the invariant distribution D^x^tc) in the following form Di(xAtc)- D(x 0 (t c ),r,p 0 (t c ),p,t c )6(xO-xO(t c ))5(pO-pO(t c )), x°(tc) = bt c ,

(13.24)

pO(tc) = Vp 2 (tc)+m 2 .

It follows from (13.23) and (13.24) that — +v-— dtc

+p*

8T

=0,

(13.25)

dp

because (a/at c ) 5(x°-x°(t c )) 5(p°-p°(t c ))=0 and D [ V ^ — + F ^ —]5(x°-xO(tc))5(p 0 -p 0 (t c ))

= D[xo

=

-{\k°

^ - + p o _ L . ] 8(xO-xO(tc))8(pO-pO(tc)) ax u

dp"

^T+ P°-^l axu apu

5

} 5(x 0 -x°(t c ))5(pO-pO(t c )).

(13.26)

In the nonrelativistic case, we have lp(t c )l = ^nvl « p°(t c ) «• m and C »»b - c in the frame F, so that J djx = d 3 xd 3 p , [x°p°] (13.27) /Di(x A ,t c )5(x° - x°(t c ))8(p° - p°(tc))dx°dp 0

-D^xMc)! „

0

x°=bt c ;p°=p u (t c )

180

=D(r,p,t c ).

A Broader View of Relativity

The last expression in (13.27) is the one-particle six-dimensional [x-space ( r , p ) , which satisfies eq. distribution D(r,p,t c ) is the probability density common time t c near the point ( r , p ) in the F frame

distribution in the classical (13.25). The non-relativistic of finding the particle at in which C=b=c.

A general 4-dimensional kinetic equation for a gas not in equilibrium is given by

( JatL + v f — + F H — ) D 1 = P ( D 1 ) , c ax^ dp*

D^D^xA.tc),

F ^ = ^ , dt c

(13.28)

where P(Di) is given by Boltzmann's collision postulate (i.e. the variation of Di per unit time due to collisions, see eq. (13.78) below) and it should fulfill certain requirements. 5 Once the solution of the kinetic equation (13.28) is obtained, one can compute the energy-momentum tensor of the relativistic fluid THv(x) = /d 4 p2e(p°)8(p^ 2 -m 2 )p^p v D 1 (x A ),

pH=^d^H=Gv^.

(13.29)

ds The average value of a measurable quantity Qix,p) is defined as = Jojx,p)D 1 (x,p)du,

xA = (x,p).

(13.30)

The particle current density ji*(x) in the 4-dimensional spacetime can be defined in terms of j^(x,p)=Di(x,p)vH- in (13.20) with A=0,l,2,3, as follows, jf(x) = /d 4 p2G8(p°)6(pk 2 -m 2 )Di(x,p)vi\

GvM- = pi*.

(13.31)

Also, the invariant density of a fluid can be defined as /D 1 (x,p)6(p^ 2 -m 2 )2Ge(pO)d4p .

(13.32)

This can be verified in the simple case where D!(x,p) = Ap(x)exp(G/t). In the presence of the external electromagnetic field tensor f^v,

Common Relativity and Many-Body Systems

when

181

P(D!)=0 and aD^at^O, (13.29), (13.28), (13.29) and GvH = p(* lead to dvJ^v = jlx(x)^,

(13.33)

where j(i(x) is given by (13.31). Clearly, the result (13.33) reduces to the conservation law a^1V-v=0 when f**v= 0. 13 d.

Invariant Liouville equation

We consider the Gibbs ensemble of similar systems of N particles, i.e. identical in composition and macroscopic conditions, but in different states. (cf., section (13.e) below) Such an ensemble is represented by an invariant Nparticle density function DN(xa\c)=pi(a.Pa>tc)> a=l,2,...N, which is normalized by the relation

/p,(xa,Pa.tc)dr =

p* fjd4xbd4Pb = N,

(xa,pa) = (x£,p£) ,

P*-Pt(xa,Pa,tc) ft 5(t c -Xoa/ba)6(pI-m a )e(p2)(2G a ),

(13.34)

(13.35)

where dr is given by (13.16). The density p satisfies the continuity equation

—P.+ — ( V S P I ) + — ( P S P I ) - O ,

atc '

dx»

(13.36)

apg

where one sums over both a=l,2,...,N and ^=0,1,2,3. With the help of the Dirac Hamiltonian and eq. (13.13), we are able to obtain one single 4-dimensional invariant Liouville equation in common relativity, ( ^ T + V a — +Fa — )Pi-^-Pi"°. 3t dt c axg ap£ c

<13-37>

for a system of N particles. We note that, in the framework of special relativity,

182

A Broader View of Relativity

one can only obtain N one-particle

Liouville equations 2 rather than one single

invariant Liouville equation:

( axb

+ —&—^-+ -—b. )pi(xb,Pb,tb) - -—Pi(xb,Pb-tb) = 0 , dx b S x £ dx b a p £ dx b b is not summed,

(13.38)

b=l,2,...N ,

where xb is the proper time of the particle b. The basic reason for this property is the absence of a universal (or common) time for all observers in special relativity. As a result, within the framework of special relativity, there is no notion of the canonical evolution of an N-particle system and the entire history of the system of N particles is represented by an N-dimensional manifold r a t h e r than a one-dimensional manifold. With the help of (13.35) and the relation (13.26) for each particle, the Liouville equation (13.37) takes the more familiar form ( — + 2 ^ ' — v atc ^ ar a

+

S V ^

—)PaPa'

0

'

(13-39)

p/xa.Pa.tc) - P ( x a ( t c ) , r a , p g ( t c ) , p a , t c ) n 5(xg - xg(t c ))6(pg - pg(t c )) , b=l 0/t i

xa(tc)=batc,

Pa(tc) = VPa(t c )+m 2 ,

Ga= ^

,

(a is not summed).

Thus, we see that the explicit four-dimensional feature of (13.37) has been lost in this case, and the 3-dimensional form in (13.39) is the same as the usual Liouville equation in the nonrelativistic limit. The ensemble average of a measurable property f(xa,Pa) of a system is defined by / =

f(x a ,p a )p I (x a ,p a )dr r-. — JpiXa.PaJdr

•

Common Relativity and Many-Body Systems

(11-40)

183

Relations poa=*yp|+m| and xoa^>atc are understood in the functions p(xa,pa,tc) and f(xa,pa) in (13.40). 13e. Invariant entropy, temperature and the Maxwell-Boltzmann distribution Let us consider the distribution of momenta of a relativistic gas at equilibrium in a box having the invariant volume (13.2). Suppose one divides all the quantum states of an individual particle of the gas into groups denoted by i=l,2,3,... Each group contains neighboring states (with approximately the same genergies). Both the number of states Wj in group i and the number of particles n; in these states are very large. Then the set of numbers nj completely describes the macroscopic state of the gas. Each particle in group i has an invariant genergy Gi. These quantities satisfy the conditions Jni-N, i

2niGi=Gt°t-

(13-41)

i

Putting each of the n; particles into one of the w; states, one obtains (Wj)ni possible distributions. Since all particles are identical, these possible distributions contain some identical ones which differ only by a permutation of the particles. The number of permutations of n-, particles is n;!. Thus the statistical weight of the distribution of nj particles over Wj states is (Wj)ni/ni!. We define the entropy Sc of the gas as a whole to be

Sc = ^lnAr i (

AI\ = (wj)"' /(nj!).

(13.42)

i

When the gas is in the equilibrium state, the entropy of the system must be a maximum. Suppose the average number of particles in each of the quantum states of the group i is denoted by , i.e., =ni/wi. One can find those , which give Sc in (13.42) its maximum possible value subject to the conditions in (13.41). By the usual methods of maximization of Sc and the Lagrange multipliers, we obtain the most probable distribution

184

A Broader View of Relativity

< n > = A expt-Gj/x],

(13.43)

where A is a normalization constant and x is a "scalar temperature." that is an invariant function and can be used to define

We stress invariant

thermodynamic quantities. 3 (For a derivation of (13.43) using the genergy, see (13.95).) Similarly, we can obtain the invariant Maxwell-Boltzmann

equilibrium

distribution (normalized by (13.17)) D^ B (p) = BN exp[-G/x],

(13.44)

with

Jp v

2

+m2 —,

„ 1 2x, v fm B = —pm^Vi-ccKT — 4 lex

where Kv (z) is the Bessel function with an imaginary argument and can be expressed in terms of the Hankel function

H^(iz):

K v (z)=(jii/2)exp[-i3tv/2]

xH^Hiz). In the nonrelativistic limit, we choose the frame F in which C=c and make the approximation of large mass m or Vp 2 +m 2 =*m+mv2/2c2. We have nMB I ""

N exp[-m/cx-p2/2mct] 7

I

Vi4mzcxVn;cx / 2m exp[-m / ex] =

N / V

'

(2jimc)

3/2

exp[-mv 2 /2c 3 x].

(13.45)

Comparing this with the Maxwell-Boltzmann distribution °<:exp[-mv2/2kBT], we see that the invariant temperature x is related to the usual temperature T and the Boltzmann constant kg by the relation (in the F frame): x = kBT/c3.

(13.46)

Note that the concept of an invariant temperature x is closely related to the fact that the genergy G of a particle is invariant and proportional to the energy in

Common Relativity and Many-Body Systems

185

the F frame. In special relativity, the usual procedure is to introduce an inverse t e m p e r a t u r e 4-vector for the invariant

form

of the

Maxwell-Boltzmann

distribution. However, it is not at all clear that such a notion of a temperature 4-vector has any physical meaning.2 13f.

Invariant

Boltzmann-Vlasov

equation

For a non-equilibrium gas, one may describe the system in terms of the coordinates x^ and velocities v(* of the particles which comprise it. Suppose the system is described by the distribution f(x^,v^).

The rate of change of the

distribution f(xH-,v(*) in a general inertial frame is given by the invariant equation df dx^1 df dt c " dtc dx* ii <^ \ x^ = (btc,x,y,z) ,

dv1" df dt c dvv dx^

u af u af v1* + ar dx^ dv^

M ,„ . = v^=(C,v),

dv^

dtc

(13.47)

M , n \ = dr =(ar,a),

dt c

c d(bt r ) C = ——&-, dt^

where the velocities v^ and accelerations at4 are 4-vectors in common relativity because the common time tc is a scalar. The physical meaning of the distribution f(xM-.vM') can be seen from the following invariant relations ff(x ,l ,v |1 )d 4 v6(s 2 -C 2 +v 2 ) =n(x^) = n(bt c ,x,y,z),

s2 J d l f J d w ] 2 J d r _ ] 2 dt c J \dt c J ^dt c J

=c2

_v2>

(13.48)

dw _ d(btc) = dtc dt c

and

Cj

where n(btc,x,y,z) is the number of molecules per unit volume at the time t c and position r=(x,y,z) in a general inertial frame.

The function 5(s 2 -C 2 +v 2 ) is an

invariant constraint of the 4-dimensional velocity integration 2

f d 4 v.

This is

2

the same as a mass-shell constraint, 8(p x -m ), in the momentum integration J d 4 p.

In the framework of common relativity, we have, by definition, the

simple familiar form of the density function n(t,x,y,z) of molecules because

186

A Broader View of Relativity

b=C=c=constant in the frame F. In a general inertial frame, integrating over the invariant volume given by (13.1) yields the total number N of molecules, jn(x^)dVi=N,

(13.49)

which is a constant, independent of the common time t c and the reference frame. If the molecules do not collide, then df/dtc = 0, i.e., v(*

JL + av JL

= o,

f = f (xi\ v^).

(13.50)

This is a special case of the Liouville theorem. If the molecules collide with each other, then df/dt c is not equal to zero and the function f can be understood based on Boltzmann's collision postulate.5 For a high temperature plasma, let us consider the generalized 4-dimensional Boltzmann-Vlasov equation based on common relativity. Vlasov first applied the equation to plasmas. The equation in a general inertial frame is assumed to have the form of (13.50) with the acceleration alx=(a°,a) of charged particles expressed in terms of the electromagnetic fields: C-^—+ v • — + a°(E, B, v) — + a(E, B, v) • — = 0, d(btc) ar eC av

(13.51)

where f=f(bt c ,r,C,v)=fi is the one-particle distribution function. The relation between the 4-acceleration a^=(a°,a) and the electromagnetic fields can be obtained from the equation of motion (13.14) with t h e 4-coordinate x^=(w,r)»(bt c ,r) and the 4-dimensional transformations in common relativity. The equation of motion of a charged particle can be written as M.efnv^, ds ds

or

&L - e f»» *±, dt c dt c

(13.52)

where x v =(bt c ,-r) and

Common Relativity and Many-Body Systems

187

m ^ = (p°,p): ds

mft

(13.53)

/f^'lf 1-P j 2

The fourth component of the equation of motion in (13.52) is not independent of the three spatial components. The three independent equations of motion can be written as -^2- = e(CE + vxB). dt r

(13.54)

Using the expression for p in (13.53) and dv(Vdtc=(dC/dtc,dv/dtc)=(a0,a), eq. (13.54) becomes my a + i2 ^-

C2

( v a ) - mY3 —

= e(CE + v x B). .2 / r 2 ift-vVC

(13.55) Now let us consider the Boltzmann-Vlasov equation in the F frame, in which the speed of light is constant, i.e., a 0 =C = 0 or C = c = constant. One can solve for the acceleration a in terms of E and B by setting a= Zi(CE + vxB) + Z2V(vE) and using (13.55) with C = 0 to determine Zi and Z2- We obtain = —(cE+vxB)-—v(v-E), my v

C = 0.

(13.56)

'my

Thus, the generalized 4-dimensional Boltzmann-Vlasov equation based on common relativity can be written as af af ce (cE + v x B ) - — v ( v - E ) •—= 0, — + v — + my my at r ar av

(13.57)

in the F frame. If the second order terms in (v/c) are neglected, equation (13.57) reduces to the usual Boltzmann-Vlasov equation,

188

A Broader View of Relativity

af af r e L v x B ] af „ — + v— + E+ — = 0. atc j r mi c sv

(13.58)

Let us consider another inertial frame F' in which the speed of light c' is not a constant, C=cV0.

In this case, (13.55) is written in terms of primed

quantities measured by observers in the F' frame, my c' + Y ' 2 ^-(v'<j') -my ^ v'a'° , 3 ^ - = e(c'E'+v'xB'). c' c' z

(13.59)

The quantity a'o= c' in F' can be solved by multiplying (13.59) with v'. We obtain c

an = — v a R'2

e -(v'E') my''

C

(13.60) C'

T/M

Substituting (13.60) into (13.59), one has my c'

a'-t

p .2 c ,2

, , (v'tf) - - ^ - ( E ' v ) = e(c'E'+v xB'). p , 2 c'

(13.61)

One can solve for the acceleration a' in terms of E' and B' by letting a' = Yi(c'E' + v'xB') + Y2V'(v'-E') and using (13.61) to determine Yi and Y2. One finds that a

'0 = 'j + Y2W'-E', and [my' ' (13.62) ,

,

a ' = — c E'+v'xB +Y2—(E'V) my' I c' where the function Y2 cannot be determined because there are four u n k n o w n variables, but only three independent equations of motion.

However, the

4-acceleration aV- in F and a'^ in F' must be related by the

following

4-dimensional transformation: ao=Y(a'o+P a 'x).

a x =Y(a'x+Pa'o)>

ay=a'y,

az=dz.

Common Relativity and Many-Body Systems

(13.63)

189

For simplicity, let us consider the y-component in (13.63). From (13.56) and (13.62), we have ce (CE + vxB) + — v v (v-E) v 'y my ' my < — (c'E'+v' x BJ, ) v +Y 2 —ME'-v') my' I c'

(13.64)

Based on the transformations for E, B and vf* in equations (13.33) and (13.9), one can show that ^ = -LVc2-v2=^-Vc'2-v'2=^ , my m m my' (13.65) (cE + vxB)y = (c'E' + v'xB')y ,

and

vy=vy

It follows from (13.64) and (13.65) that

Y2=-

ce (E-v)/c my (E'-v')/c'

(13.66)

If one wishes, one may express c/y and (E-v)/c in (13.66) in terms of quantities measured by observers in the F' frame with the help of the transformation properties of (E-v)/c (or the 4-vector in (13.73) below) and the invariant Vc^-v 2 =Vc'2-v'2. Note that this function Y2 can also be obtained from other relations in the transformation (13.63) for the 4-acceleration at* in common relativity. Thus, the 4-acceleration a'v- is completely determined by (13.62) and (13.66) . c . .(vE) a'0- my1 v'-E - c (13.67) a' =

190

c'E'+v'xB'-v'^^ my

A Broader View of Relativity

Thus, the generalized Boltzmann-Vlasov equation (13.51) in the F' frame can be written as af

, af

ec'

— + v •— + — V - E ' - c ' ^ atc ar' my'

ed

my' where f'=f(b't c ,r',c',v').

af

c'E'+v'xB'-v'^^

av

(13.68)

Since (v-E,cE+vxB) and ( c , v ) are 4-vectors, the

quantity (v-E)/c in (13.68) can be expressed in terms of the primed quantities v', E', B' and c' measured by observers in F' by using transformations (13.9) and (13.33). Nevertheless, it is convenient to leave the unprimed quantity in (13.67) and (13.68). Using the velocity transformation (13.9), one can also show that the accelerations a^=(0,a) in (13.56) and a'l* in (13.67) are related by a 4-dimensional transformation. Combining (13.56) and (13.67) gives

a?u a 2 = - — (v c E + v x B ) - — v ( v - E ) lmy ' my I

(cE + v x B ) 2 - ( v - E )

2 (v-E) 2

(13.69)

where ao=0, y ^ l / v l - v ^ / c ^ , and n2

0

my'

ce my

v'E'-c'—

,(v-E) (c'E'+v'xB'j-v'

2

- -.E.)2_(V^ (c'E'+v'xB') -(v'-E')

(13.70)

where we have used the invariant relation c/y = c'/y'. Note that the unprimed terms (v-E) 2 /y 2 in (13.69) and (13.70) are the same. Thus, the invariant relation

Common Relativity and Many-Body Systems

191

„2

„2

„2

„,2

„,2

(13.71)

is equivalent to the invariant relation (cE + v x B ) 2 - ( v E ) 2 = (c'E' + v'xB') 2 - (v'-E') 2 .

(13.72)

This is not surprising because ^

=

ef^v

^ L

=

e/ C v . E ) , (cE + v x B))

(13.73)

dt

is a 4-vector in common relativity. It is stressed that the acceleration is the quantity with the simplest physical properties in common relativity because of the presence of the common time t c and the 4-dimensional symmetry. 13g. Boltzmann's

transport

equation

with

4-dimensional

symmetry For the classical kinetic theory of dilute gases for a system of N molecules enclosed in a box of volume Vi, we are interested in the invariant distribution function f under a 4-dimensional transformation.

Let us define the distribution

f as a function of the coordinates and momenta, (xl i ,p v )s(x,p), instead of a function of the coordinates and velocities (x^,v v ). Such a definition of f(x,p) will make the 4-dimensional symmetry more explicit, as we shall see below. In a general inertial frame, the invariant volume element in the n-space given by (13.15) can be rewritten as

dn = d x8

d 4 P 8(Px-m 2 )9(p°)(2G)

• d 3 xdx°5|^ - t

u ,0*,„2 d- 5 pdp 6(pf-m^)e(p u )

2p°

The distribution function f(x,p) is defined so that classically, the number of molecules dN at common time tc in a volume element d 3 x about r and within a

192

A Broader View of Relativity

momentum space volume element d 3 p around p is given by f f (x, p)dn = j f (x, p)dVjd3p ^-,

dN = dN =

(13.74)

[x°p°]

dV,=

d4x

/

(13.75)

[x»]

d

3

p ^ = ^ P (2Gc 0 ),

C

2 p°

p°=T/p 2 + m 2 ,

(13.76)

where integrations over x° and p° have been carried out in (13.74) and Co is a scaling constant defined in (13.1) to give the correct dimensions for dVi. Note that both dVi and d3p(c0/C) are invariant in common relativity and that they are large enough so that dN is a very large number (~1010) of molecules. Nevertheless, the volume elements dVi and d 3 p(c 0 /C) are microscopic in size. Also, if the distribution f(x,p) is independent of the position r, we have the usual relation, f f(x,p)d3p=N/V, in the inertial frame F(ctc,r) in which c0=c=C and f dV; =Vi , where Vi is the volume of a box. The distribution f(x,p) can be identified with Di in (13.28) when Di does not depend on t c explicitly, f(x,p)=Di(x,p). Let us consider equation (13.28) with the collision term P(D), where we use D to denote the one-particle distribution function for the following discussions, d

u

+ V^

^tc

d 3X ^

dx"

dt r

d

+r Pl i

D = P(D),

D = D(x,p,t c ),

(13.77)

dp")

= (Qv),

dp*1 F^ = dt r

d p 0 dp dt c ' dt c

where the collision term P(D) = (flD/at c ) co u must be specified for (13.77) to be meaningful.

In order to have an invariant expression for the collision term

P(D), we use quantum mechanics to treat the scattering process. Although the molecules are regarded as classical objects, "they see each other as plane waves

Common Relativity and Many-Body Systems

193

of definite momenta rather than wave packets of well-defined positions''^ in scattering processes. By considering the collision process, Pi+P2-*Pj +P'2> o n e can express (aD/at c ) c o n =(aD 1 /at c ) c o n in the invariant form,

P(Dl) = K ) flt

I c icon

=2 / ^PifPLfPi 6 4( Pl+P2 _ Pi _ p .;2 ) 2 v p 2 0 Pio 2p' 20

x|Mfi| {p'{D'2-DiD2),

(13.78)

where Di=D(x,pi,tc) and D'i=D(x,pl,t c ), i=l,2. From equations (13.77) and (13.78), we have the 4-dimensional Boltzmann transport equation based on common relativity: '

d

n

atc

d

fl(btc)

o

a

dp,

s

ar

dt c

flpo

=;^22-^TL^rk-54(p1 2P20 2pio 2p 2 0

+Pl

dpi

a

dt c apj

D,

- p i -p'2)|Mfi|2(DiD'2 -EhDz). ' '

(13.79)

In the inertial frame F(ctc,r) in which b=c=C, if the distribution function Di is a function of x and p o n l y / then the left side of (13.79) reduces to the more familiar form aDi aDi dp° aDi dpi aDi —*-+vr—L+ ^ i . + -ZL L ^ ar dt^ a p o dt c dpl

„ „ , , D 1 =D 1 (x,p).

,-,-,on\ 13.80)

Note that the third and fourth terms are not independent because of the relation Pj^Pj+m2.

However, it is more convenient to leave Boltzmann's transport

equation in the form (13.80) [or the left side of (13.79)] to have an explicit 4dimensional form. We also note that (dpi°/dt c ) is smaller than ldpi/dt c l by a factor v/c, as one can see from (13.73). Thus, if the velocity v=lvl is much smaller than c, (13.80) reduces to the usual form: aDL+pI.aDL+dpL.aDI atc m ar at c apj

194

A Broader View of Relativity

In this connection, it is worthwhile to note that if the Boltzmann-Vlasov equation for high temperature plasma is expressed in terms of the distribution function Di(x,p), we have p aDi aDi + -*— •—L a(btAar c) po

-EpaDi po dpo

_

pxB

+ e———L+e

(13.82)

dp

in a general inertial frame, where we have used the following relations 1 1 1 PO

\V

<

m

m

i

2

Vl-v /c 1 dp0 _ eEv C dt c C

13h. Boltzmann's

mv/C

r

2

>

2

Vl-v /C 1 dp Cdtc

_Ep p0 '

H-theorem

with

2

v C

p Po

_n PxB E + —— v v i

4-dimensional

(13.83)

(13.84)

symmetry

In common relativity, the invariant Boltzmann functional H(t c ) can be defined in terms of the invariant distribution D(p^,t c )sD(p,tc),

H(tc) = / D ( p 1 , t c ) l n D ( p 1 ) t c ) ^ L ,

pio = J p 2 + m 2 ,

(13.85)

where the invariant distribution D is assumed to be a function of the 4momentum
^ - / ^ | ^ | ^ S 4 ( P l atc •> 2p 2 0 2pi 0 2p' 20 <Mfj (DiD^-DjE^),

+ P2

- P i -P2)

D^Dfp^tc).

(13.86)

Note that both the distribution Di and d 3 p 1 / ( 2 p 1 0 ) are invariant. It follows from (13.85) and (13.86) that

Common Relativity and Many-Body Systems

195

rf3r,, H^iV. H^w. dH(tic)) _ f dl3<-,. f a d ^ d V i dV2 6 4 ( P l + P 2 - p i - P - 2 ) dt •J 2?n-i^ ?non ?riirt0 2p' ?n'-ir» c p 1 0 2p 2 o 2pi 20

x|Mfi| (DiD^-DiDz^l + lnD!].

(13.87)

Note that the transition matrix Mf j is invariant under the interchange of px and p2, so that the integrand in (13.87) is unchanged under pj •* p 2 . From (13.87) and the new expression obtained by pi *» p 2 , we have

dtc

J

2pio 2p 2 0 2pio 2p' 20

xM fi (DiD' 2 -D!D 2 ) l + ilnCDiDz)

(13.88)

For every scattering there is an inverse scattering with the same transition matrix Mfi. Thus, the integral (13.88) is invariant under the i n t e r c h a n g e IP1.P2l-iPi.P2l.

dHOJ

r d ^ d ' p z d V x dV 2 J

d^

64(pi+p2_p,

_p,}

2pio 2p 2 0 2pio 2p' 20

xM fi (DiDj-DxDz) l + |ln(DiE>2)

(13.89)

It follows from (13.88) and (13.89) that

d ^ ^ ^ Jd i a d ^ d V , ^ ^ dt c

4

2p 1 0 2p 2 0 2pio 2p' 20

x|Mfi|2(DiD'2 - D1D2)[ln(D1D2) - ln(DiD 2 )].

(13.90)

Since the integrand of this equation is non-negative, we have dH(tc) s0 dt r

196

(13.91)

A Broader View of Relativity

in all inertial frames. We observe that equation (13.91) vanishes, i.e., dH(tc)/dtc=0, if and only if the integrand of (13.90) vanishes identically. dH(t c )/dt c =0 is the same as

In other words, the condition

Db(Pi)Do(p'2) = D 0 ( P l )D 0 (p 2 ),

(13.92)

where D0(p) denotes the equilibrium distribution.

This has an interesting

implication concerning the explicit form of the invariant equilibrium (or Maxwell-Boltzmann) distribution D 0 (p). Equation (13.92) leads to lnD' 0 (pi) + lnD' 0 (p' 2 ) = l n D o ( P l ) + lnD 0 (p 2 ), for any possible collision process, P i + p 2 - * P i + p ' 2 .

(13.93) Thus, the relation (13.93)

can be interpreted as a conservation law. Since the function lnD 0 (p) is a scalar in common relativity, the only quantity which is both a scalar and conserved is the genergy G(p). In the F frame, in which C=c, we have G(p)=po/c=VP 2 + m 2 / c . Clearly, the conservation of the genergy, G( P l ) + G(p2) = G(pi) + G(p 2 ),

(13.94)

is the same as the conservation of the energy po in the F frame.

Since the

genergy G(pi)=pio/Ci is an invariant, the relation (13.94) holds for all inertial frames. Thus, the only solution of (13.93) is lnD 0 (p) = -AG(p) + lnB,

or

lnD 0 (p) = Bexp[-AG(p)],

(13.95)

where A and B are arbitrary constants that can be determined by measurements of the physical properties of a system. In the literature, most of the formalisms on relativistic mechanics are not manifestly covariant.

statistical

Some of them are, strictly speaking,

not covariant because of the approximations involved.

Some formalisms lack

proof of their effective covariance.2 We have demonstrated some novel and interesting

features of the

4-dimensional symmetry framework of common relativity. This framework has

Common Relativity and Many-Body Systems

197

a considerable advantage over the usual spacetime framework with regard to the formulation of statistical mechanics within a 4-dimensional symmetry framework. The existence of a scalar common time to the 4-coordinate (btc,x,y,z), and the genergy G in common relativity is the key to preserving the invariant phase space of initial particle positions and velocities, the concept of a Hamiltonian system with many degrees of freedom, the Liouville equation and the invariant Maxwell-Boltzmann distribution with a scalar temperature.

198

A Broader View of Relativity

References 1.

See, for example, V. Fock, The Theory of Space Time and Gravitation

(trans.

2.

R. Hakim, J. Math. Phys. 8_, 1315 (1967); J. L. Synge, The Relativistic

3.

J. P. Hsu, Nuovo Cimento B, 80, 201 (1984); J. P. Hsu and T. Y. Shi, Phys. Rev.

by N. Kemmer, Pergamon Press, London, 1958), p. 80. Gas

(North-Holland, Amsterdam, 1957), p. v. D. 26, 2745 (1982) and references therein. 4.

P. G. Bergman, Phys. Rev. 84, 1026 (1951).

5.

D.C. Montgomery and D. A. Tidman, Plasma Kinetic

Theory

(McGraw-Hill,

N.Y., 1964), P. 85. 6.

K. Huang, Statistical

7.

1987), pp. 60-62. In the case where Di43i(x,p), the 4-dimensional transport equation (13.79)

Mechanics

(2nd ed., John Wiley & Sons, New York,

or (13.80) can be used to derive the conservation theorem associated with the distribution function (see ref. 5). For example, one can show that

/PzCx,p)f^l J 2PO

latcJcou

=0,

and

/!£*"

v^

_i_ + F^ -L D(x,p) - 0, 3X M-

apH-

where (flD/dt c )coll is given by (13.78), and Z(x,p) denotes the conserved quantity associated with the collision process

pj +pj -* Pi +P2,

Z(x,p1) + Z(x,p 2 )- Z(x,pi) + Z(x,p'2). These results are useful

for

i-e.,

deriving

conservation laws of mass, momentum, etc. in hydrodynamics in a 4dimensional symmetry framework. This is equivalent to assuming that the distribution function D(pi,t c ) in a general inertial frame (1) is independent of the 4-coordinate xM^ and (2) satisfies Boltzmann's transport equation (13.28) without an external force, F^=0, and with the collision term P(Di) given by (13.78).

Common Relativity and Many-Body Systems

199

14. Common Relativity and the 3K Cosmic Microwave Background 14a. Implications of an invariant and non-invariant Planck's law for blackbody radiation Since the anisotropy of the 3K cosmic background radiation was experimentally established, 1 many physicists have concluded that it is now possible to talk about "absolute motion".2 This conclusion was reached within the framework of special relativity, in which the Planck law for blackbody radiation is not invariant under the Lorentz transformation. 3 If one is not aware that the 4-dimensional symmetry can be understood from a broader viewpoint than that of just special relativity, this conclusion appears to be unavoidable. However, for Planck's law for blackbody radiation not to be invariant is unnatural and inconsistent with the fundamental PoincareEinstein principle of relativity for physical laws. As we have shown in previous chapters, nature can be viewed from the standpoint of a 4-dimensional theory of relativity with a common (but not absolute) time for all observers. Existing experiments cannot distinguish between common relativity and special relativity because they both have the 4dimensional symmetry of the Lorentz and the Poincare groups. Nevertheless, the conceptual framework of common relativity is not exactly the same as that of special relativity. For example, we can introduce the concepts of an invariant genergy G=po/C and the invariant volume of a box for a many-particle system. With the help of these concepts, we can formulate a new covariant thermodynamics which leads to an invariant Planck law.4 14b. Invariant

partition

function

Let us consider systems with discrete states labelled by an index i, since the generalization to systems with continuous states is straightforward. Each

200

state of the system corresponds to an invariant genergy G=p°/C > 0.

Because of

its in variance, the genergy G corresponds more closely to the scalar energy i n classical physics than the non-scalar energy p°

We assume, as usual, that the

interaction between molecules (or particles) is sufficiently weak so that there is an exchange of genergy between molecules but no change in the structure and properties of a molecule. Consider a gas with N particles, where N»l. Suppose the \i space volume is divided into K cells with K»l. Each particle in the cell a>j is in the state i and has genergy Gj. Also suppose there are n; particles in the cell co;. Assuming that the a priori probability of the cell o>i to be occupied is 1, we have the probability

W(n!,n 2 ,...) = N!/

(14.1)

rini! \ i

/

for n i particles in coi, n2 particles in a>2, etc. The equilibrium state of the gas corresponds to a state of maximum W(n 1 ,n 2 ,...) consistent with the constraints

i

Gt=^niGi=NG,

(14.3)

where G is the average genergy of the constituent particles. (For simplicity, we do not consider modifications due to the radical length.)

In equilibrium, we

have nj = h~i, where

-^

=

Nexp[ p G ^

p = parameteri

(144)

which is obtained by varying W(n 1 ,n 2 ,...) with constraints (14.2) and (14.3). The invariant partition function Z is defined by

Common Relativity and the 3K Cosmic Microwave Background

201

Z = ^exp[-pGi],

p>0,

(14.5)

i

provided the summation over the states i does not change the invariance of (14.5). From (14.3) and (14.5), one has G = - a l n Z / a p . Once the partition function Z is known, all other

thermodynamic

quantities can now be derived. We can define an invariant "free genergy" F c so that Z = exp[-p~Fc].

(14.6)

We note that (14.5) is the invariant partition function of the Gibbs ensemble, in which there is no exchange of momentum, within the framework of common relativity.

This can be compared with the conventional Lorentz-

invariant partition function, in which one must introduce a 4-vector of the inverse temperature whose operational meaning is, in general, not at all clear. In a particular frame F, the partition function in (14.5) can be identified with the usual partition function of the Gibbs ensemble 2jexp(-Ei/k B T] ; and the invariant parameter p in (14.5) can be related to the Boltzmann constant kB = 1.38x10"16 erg/deg K and the usual absolute temperature T. Since Gi <= m / c + Pi 2 /2mc, pi « mvj/c, and E i = m v i 2 / 2 in the F frame, we have

l / p = kBT/c3ssT.

(14.7)

We will call the invariant quantity x the "common temperature". The dimension of T is the same as that of genergy, rather than that of energy. Note that the usual temperature T is closely related to the energy E; and that neither T nor Ei are invariants in common relativity. 14c.

Covariant

thermodynamics

In conventional thermodynamics, the relation between the entropy S, the heat Q, and the absolute temperature T is given by

202

A Broader View of Relativity

dS =

(dQ )r

- ev T

( 1 4 8 )

This suggests that we relate the "common entropy" S c and "common heat" Q_c by the relation 4

dSc =

(dQx)rev

(149)

in the covariant thermodynamics formulated on the basis of common relativity. Relations (14.8) and (14.9) are the same in the frame F in which the speed of light has been defined to be isotropic, provided that Sc=S/k.B and Q_c= Q/c 3 . I n general, the definitions of S c and Q_c must be consistent with (14.9) and the Boltzmann equation

Sc = In W(n 1 ) n 2 , ...) = — . kR

(14.10)

Let us consider a change of the genergy of a closed system. It is given by the invariant equation N8G = ^(n i 5G 1 +G i 6h i ],

(14.11)

where the first term is always considered to be the "work" and the second term to be the "heat." Thus we can define the common heat Qc as

N(8Q.c)rev=^Gi5ni,

(14.12)

i

where the reversibility of the change is achieved by letting 6 rTj correspond to a change in which the probability W defined in (14.1) remains a maximum. We see that the common heat Q.c is the change of genergy of the whole system resulting from a change in the statistical arrangement of its components.

Common Relativity and the 3K Cosmic Microwave Background

On

203

the other hand, "common work" is due to the change of genergy of every component of a system without changing their statistical arrangement.5 The reversible change of hi in (14.12) can be defined as

ffi. = -N^-^6Zj,Szj,

(14.13)

because Z in (14.5) can be considered to be a function of Zj = pG; and (14.4) can also be written as 6ni=-N^2^.

(14.14)

dZi

It follows from (14.13) and (14.12) that

-(«lc) T

\

a 2 lnZ —Sz^j1^.

Wrev

^

(14.15)

> dZjaZj

where Gj = Zj/p = TZj. Thus (8Q. c ) rev /x can be represented by a Pfaffian in the variable zj and it is equal to dSc according to (14.9). After integration, we find the average common entropy Sc of a particle in the ensemble:

Sc=— + lnZ,

(14.16)

x where the constant of integration is neglected. Using (14.2)-(14.5) and (14.16), one can verify that S c =NS c =lnW = - 2 exPt-PGOiJexpt-pGQ)

where we have used the Stirling approximation: ln(N!) = N(lnN - 1).

204

A Broader View of Relativity

(1417)

For a thermodynamic system, the invariant entropy Sc can be expressed as a function of N, Gt = NG

and N/j. The equations for thermodynamic

equilibrium are invariant:

-= % , x G U JV,N

^ =PH x

,

(14.19)

,

(14.20)

UVJG,N

HE. =

x

(14.18)

fe

13N J GV

where G - Gt and V - Vi. The invariant pressure Pc and chemical potential \ic are related to the usual pressure P and chemical potential |x by the relations P

P c = -—y,-

u

lMr * c=- 3—-

3 C

(14

"21)

3 C

in the F frame. One can verify that in the F frame, the present formalism reduces to the usual thermodynamics at low energies. 14d. The canonical distribution and blackbody radiation Because of the existence of the new invariant genergy G in (13.8) and the invariant volume Vi in (13.1), we are able to define a meaningful invariant density for particles. One cannot do this within the framework of special relativity because of the lack of an invariant genergy. The number of states available to a particle in dVi can be written as

dr = 2Gc0dVI5(p£ - m2)8(p°)d4p —!— (2*J)3 (14.22)

Common Relativity and the 3K Cosmic Microwave Background

205

_JZU

J = 3 . 5 1 7 7 x l (- r3 "3 g, - c m

in an arbitrary frame, where p>, and maO are the 4-momentum and the mass of the particle, respectively.

The invariance of d r in (14.22) ensures that the

summation over the states of a system (see eq. (14.5) does not change invariance of the expression.

the

In the particular frame F, we have C=c=c0 and

hence

dr=cPxcPp 3

(in the F frame),

(14.23)

(2:tJ) '

as expected. For an ideal gas containing N particles with the same mass maO, we have the invariant canonical (or Gibbs) distribution

N(2itJ)J

dp--

exp—

dr,

(14.24)

4jim^TViCoK 2 (m / TCQ)

m2K2|

Jdpp 2 exp ^

1= ™o)

xc0

0

nr

/-Jp^-m2 p0dp0exp TC

00

2 CT]

as

(14.25)

TC

TC 0

m -»0.

For blackbody radiation, we have the partition function Z

Z = 2exp[-(Gn/T)],

206

Gn = nG

njk 0

A Broader View of Relativity

(14.26)

where the wave 4-vector kH=(ko, k)

pf

=jk^,

k

"" |

is

related to the momentum p^ by

(14.27)

, k

Note that for a massless particle, G is still invariant under the space-lightime transformation: G

k'p

Y (k 0 -pk x )

J

c'

Y(C-PV X )

k0

(14.28)

C '

because k x =k 0 cose and v x =c cose. It follows from (14.26) that

< n >= — y n exp(-nG / x) = , Z~ exp(G/x)-l which is the invariant

(14.29)

Planck distribution for blackbody radiation in the

framework of common relativity.

Comparing equation (14.29) with the usual

expression =l/(exp[/zk 0 c/k B T]-l) in the F-frame, we obtain Jk 0 _ ftkpc cr kRT

(14.30)

which is consistent with (14.7). In the present formalism, we also have an invariant

grand-canonical

distribution, the invariant "thermodynamic potential" Q c , and the i n v a r i a n t Fermi-Dirac distribution, etc.:

H^ (14.31)

FD

e x p [ ( G - |i, ) / x ] + l '

etc.

Common Relativity and the 3K Cosmic Microwave Background

207

14e.

The question of Earth's "absolute" motion relative to the 3K cosmic

microwave

background

The 3k cosmic microwave background was detected by Penzias and Wilson in 1965. It is an important discovery for understanding the universe and is believed to be radiation left over from the early epoch of the Universe in w h i c h matter and radiation were in thermal equilibrium. The 3k background radiation involves about 10^ photons per cm 3 and this temperature is assumed to be the present temperature of the universe because the effect of hot stars is negligible when one averages their temperature over all space. The 3k cosmic radiation appears to be isotropic, having the same radiation intensity in all directions, as observed on the Earth.

However, a very small anisotropy in the cosmic

blackbody radiation has been detected and this anisotropy is usually attributed to Earth's motion relative to the background radiation. made within the

framework

This interpretation is

of special relativity with

certain

implicit

assumptions. Consider two frames F and F' (which is moving with the speed V along the x-axis of F) immersed in the blackbody radiation.

The usual Planck

distribution in the F frame is

(n) =

. exp[hv/kgT]-l

(14.32),

Observers in the F' frame, however, will detect a modified distribution,

(14.33)

'~ exp

hv'(l + Vcose'/c) kBTVl-V2/c2

which is derived from (14.32) using the special relativistic transformations , v(l-Vcose/c) v= i Vl-V2/c2

208

„, c o s e - V / c cose = , 1-Vcose/c

A Broader View of Relativity

/-,„->,,% (14.34)

and the assumptions that the Boltzmann constant kB and the temperature T are Lorentz scalars.

It follows that the spectral distribution of the energy of the

blackbody radiation is given by ,_,., . 2d 3 xd 3 k hv dE(k 0 ) = , (2K)3 exp[hv/kBT]-l

in F (14.35)

"W"U'

(2K)3

exp[hv'(l + Vcose , /c)Y/k B T]-l'

The photon energies hv and hv' in (14.35) are related by the relativistic Doppler effect (14.34). Thus, the cosmic microwave background plays the role of a new "ether" and the frame F is the privileged and absolute frame in which the law of the Planck distribution takes a particular simple form, as one can see from (14.35). According to this view, nature provides a natural reference frame, i.e., the cosmic 3k radiation represents a fixed system of coordinates in the Universe. Many physicists thus concluded that we now are justified in talking about "absolute" motion. Nevertheless, this interpretation is not in harmony with the PoincareEinstein principle of relativity for physical laws. Furthermore, this conclusion is untenable from the viewpoint of the covariant thermodynamics based on common relativity.

According to the common relativistic framework,

the

Planck formula corresponding to (14.35) in F and F' can be obtained by combining the invariant density of states (14.22) with two polarizations (i.e. 2 d r ) , the invariant Planck distribution (14.29) and the covariant

photon

"energy" Jk 0 (which has the dimension of mass):

dM(k0) -

yyo

exp[G/x]-l

,

in F, (14.36)

dM'(k' 0 )=

2drJk '° , exp[G/x]-l

in F',

Common Relativity and the 3K Cosmic Microwave Background

209

where we have used the invariant genergy in (14.28). The photon "energy" Jk 0 and Jk^, in (14.36) are related by the Doppler effect in common relativity: k'o = Y(k 0 -pk x ),

k'x = Y(k x -pk 0 ),

k'y = k y ,

k'z = k z , (14.37)

k^ =

-,k

Since the density of states (14.22) and the Planck distribution =fexp[GA]-lJ"! are invariant, the quantity dM'(kg) is covariant and transforms like kg

in

common relativity. Therefore, from this viewpoint, the observed anisotropy in the cosmic 3k radiation should not be interpreted as being caused by the motion of Earth relative to the cosmic microwave background.

This is in contrast with

the corresponding result of special relativity, in which dE'(k'o) in (14.35) does hot have a covariant form and does not transform like k 0 ' because the Planck distribution is not invariant and takes the simplest form only in the frame F. Note that the Doppler shifts of h v ' in (14.35) and Jk' 0 in (14.36) are both consistent with previous experimental results. The reason for this can be seen clearly by considering the quantities involved in the Doppler shift experiment. The relation between the frequency v=cko/2ji and v'=c'k' 0 /2it in (14.37) is quite different from that given by (14.34) in special relativity. This is because we have used common time to measure the speed of light c and c' and to define the frequency of a wave in our formalism.

However, this does not contradict

laboratory experiments testing the Doppler effect involving shifts of atomiclevel structure, lasers, etc. The reason has been discussed in chapter 10 [eqs. (10.7)-(10.18)]. Experimentally, the anisotropy of the cosmic 3k radiation was established at the level of one part in 10 3 to 10 4 .

According to common relativity, this

anisotropy should not be interpreted as being related to the angular dependence of the Planck distribution due to the motion of the Earth relative to the 3K cosmic radiation.

Instead, the anisotropy of the 3k radiation indicates a new

phenomenon on a cosmological scale. For example, it may be related to a small nonsymmetric expansion of the Universe or large-scale irregularities in the distribution of energy in the Universe.

210

Whether or not these interpretations

A Broader View of Relativity

are viable could be clarified in the future when one is able to measure the dipole anisotropy (~10~5) due to the Earth's orbital motion and the quadruple anisotropy.

Common Relativity and the 3K Cosmic Microwave Background

211

References 1.

P. J. E. Peebles and D. T. Wilkinson, Phys. Rev. 174, 2168 (1968); G. F. Smoot,

2.

V. F. Weisskopf, Am. Sci. 7 1 , 473 (1983); S. Weinberg, Gravitation

M. V. Gorenstein and R. A. Muller, Phys. Rev. Lett. 39, 898 (1977). Cosmology, Principles and Applications

of the General Theory of

and

Relativity

(Wiley, New York, N.Y.m 1972) p. 506 and reference therein. 3.

S. Weinberg, Gravitation and Cosmology, Principles

and Applications

General Theory of Relativity (Wiley, New York, N.Y. 1972) p. 522.

of the One may

introduce an inverse temperature 4-vector, etc. to make the Planck's law of blackbody radiation

invariant

under

the

Lorentz

transformations.

However, the operational meaning of such an inverse temperature 4vector is in general not well defined. 4.

J. P. Hsu, Nuovo Cimento B, 93, 178 (1986).

5.

B. Touschek, Nuovo Cimento B, 58, 295 (1968).

212

A Broader View of Relativity

15. Common Relativity and Quantum Mechanics 15a. Fuzziness at short distances and the invariant genergy One of the basic problems in quantum mechanics is the fact that the non-square-integrable coordinate representation is incompatible with the usual probabilistic interpretation. This situation suggests the need to introduce fuzzy coordinates to modify physics at short distances.1 The introduction of fuzzy coordinates is also motivated by the problem of locality and the related ultraviolet divergence in quantum field theories. Fuzzy coordinates may be significant because an inherent fuzziness at short distances is characterized by a radical length R. This radical length R is related to the fundamental length RD in quantum field theory, a concept that has was introduced many years ago by Heisenberg and Dirac.2 Such an inherent fuzziness is also related to Feynman's idea3 of a basic width for a modified 6-function for interactions. In 1949, Dirac made the following comments: "Presentday atomic theories involve the assumption of localizability, which is sufficient but is very likely too stringent ••• A less drastic assumption may be adequate, e.g., that there is a fundamental length X such that the Poisson bracket of two dynamical variables must vanish if they are localized at two points whose separation is space-like and greater than X, but need not vanish if it is less than X." Schwinger, 4 Feynman5 and Wigner6 also had similar views. As we have seen, common relativity has a new invariant G(p) = p°/C 2 =-\/p +m2 /C which is called the "genergy." In the F frame in which the speed of light is, by definition, a constant, i.e., C=c, the genergy G(p) is essentially the same as the conserved quantity, energy, p°=-\/p2 +m 2 . The invariant genergy G(p) can help us to operationalize an inherent fuzziness at short distances without upsetting the 4-dimensional symmetry. This is not possible to do in the

213

framework of special relativity because the genergy G(p) is not an invariant quantity in special relativity. The concept of the genergy G(p) makes possible a departure from the 4-dimensional symmetry of special relativity at short distances. This is but another example in which common relativity, by allowing the definition of new invariant quantities not possible in special relativity, may be more convenient than special relativity for certain types of calculations. First, let us show the invariance of the genergy G by calculations based on the 4-dimensional space-lightime transformation: 1 The transformations of the 4-momentum pi* = mdx^/ds are given by : P'° = Y(P0-PPX1

!

/

•

VTT

p' x = Y (p x ~f3p 0 ),

p'y=Py,

P'Z=PZ-

(15.D

,2

where the spatial components of the momentum are p-CPx.Py.PzMP 1 ,? 2 ,? 3 ). It follows from (15.1) and the velocity transformations (8.5) that the ratio p°/c is invariant under the 4-dimensional transformations:

£-=P-=G(p), c' c

dtc

'

p , 0 = -Jp' 2 +m 2 ,

Po

p°=-ip2+m2.

(15.2)

c dtc'

Note that the genergy G(p) involves the spatial momentum vector p .

This

invariance is quite interesting because it suggests the existence of an i n v a r i a n t function involving p and a new universal constant I which can be related to a radical (or fundamental) length.

The constant I would allow us to define a

dimensionless function D(p) in terms of the scalar genergy G(p) associated with a physical particle of mass m,

D(P) =

214

± ^ l + I 2 G 2 (p)

(15.3)

A Broader View of Relativity

The constant I has the dimension of length/(mass-time). Thus, D(p) is a pure number and may be related to a probability. The constant I can be related to a radical length R defined by R = —,

(15.4)

Co

where c0 is the scaling constant defined in (13.1). The radical length R in nature represents a basic physical quantity which is independent of the artificial choices of units for mass and time. Naturally, the numerical value of R still depends on our choice of a unit for length. However, the radical length R, if it exists, should be regarded as the unit of length chosen by nature rather than by humans. This has physical significances, as we shall see later. Note that the invariant relations (15.2) and (15.3) do not exist in special relativity. 15b. Fuzzy quantum mechanics with an inherent fuzziness in the position of a point particle In quantum mechanics and quantum field theories, basic physical objects are assumed to be point particles which may have a definite and precise position in space (or a definite momentum, but not both at the same time because of the uncertainty principle). As a result, the Coulomb potential produced by a point electron takes the form -e/(4nr), which diverges at r=0. This divergence is intimately related to the fundamental divergence difficulty in quantum electrodynamics and other field theories. In addition, the point particle picture implies that a quantum particle can have a position eigenstate Iq) of a position operator Q: QJq)=qlq>.

(15.5)

Therefore, in principle, the position q of a quantum particle can be measured precisely, i.e., Aq m i n =0.

(15.6)

Common Relativity and Quantum Mechanics

215

However, the properties (15.5) and (15.6) do not have operational meanings because they cannot be realized. Furthermore, the position eigenstate (15.5) is, strictly speaking, not physically meaningful because its wave function is not square-integrable and hence, cannot be normalized to have an elementary probabilistic interpretation.

Thus, in quantum mechanics, one gives up the

fundamental idea of a probabilistic interpretation of a state for the sake of introducing certain mathematical idealizations.

This is inconsistent with the

notion of operational definitions in physics, which states that one should formulate a physical theory using observable quantities and realizable states. On the other hand, if one attempts to avoid the above difficulties

by

assuming the particle to have a finite size, then one immediately encounters other difficult problems related to non-local interactions, n o n - l o c a l wave functions and so on. One way to depart from the conventional point particle picture without these difficulties is to assume that the particle itself has no size and no structure, but that its position cannot be measured with unlimited accuracy: 1 Aqmin=R>0,

(15.7)

where R is a very small radical length.

The postulate (15.7) enables us to avoid

unknown tacit assumptions such as indefinitely short distances in space. 5 This suggests a picture of a fuzzy-point particle, which closely resembles a fuzzy point in the fuzzy set theory of Zadeh. 7 According to postulate (15.7), we replace the improperly idealized state (ql with zero width by a new base state (ql, which has a width R. To accomplish this, it is convenient to use Klauder's continuous representation 8 of Hilbert space to express the new base (ql: (ql- ?dp(plD(p)ei'x"J-7i=, _„

(15.8)

V2JIJ

where (p I is the usual momentum eigenstate.

In order to preserve known

properties of a physical state at low momenta, the function D(p) must satisfy

216

A Broader View of Relativity

1

|p| « J / R;

0

|p| » J / R.

D(p) =

(15.9)

Similarly, we can replace the usual momentum eigenstate (pi with a momentum state (pi which has an inherent fuzziness APmin=|>0,

(15.10)

where p and S have the dimensions of mass and length, respectively. To be consistent with known physical states at low momenta, the length scale S should be extremely large. Presumably, it is related to the size of the observable physical universe S&10 1 0 light years. For simplicity, we postulate that the physical effects of a finite S are too small to be detected experimentally and work in the limit in which S approaches infinity so that (pi =
(15.11)

Instead, we will concentrate on the new physical properties implied by the radical length R>0. Consider the fuzzy base states. In the limit S^«>, the base momentum states (pi and Ip) have the usual orthogonal and completeness relationships

(plp') = 5(p-p'),

Jdplp)(pl=l, (15.12)

(plP = (plp,

(plQ. = iJ — ( p i , dp

where Q. and P are Hermitian operators for positions and momenta. Because R>0, the position vector (ql, its dual vector Iq) and the Hermitian operators Q. and P satisfy ( q l q ' ) - ( q ' l q ) ' - f ^ . D ^ e 1 ^ ' * " * = D2(d)S(q-q'),

(15.13)

J 2n]

Common Relativity and Quantum Mechanics

217

(qlP = - U A ( q | , aq

f d q I q)(q I = D 2 (p),

Plq) = i j A | q ) j aq

JdqlD-^a) I qMD^OXq I] = / d q I q)[D"2(a)(q I] = 1,

(qld-(ql[q-Ui^L aP

(15.14)

J

Q. I q) - [q + U ^ ^ ] aP

(15.15)

I q),

(15.16)

where D2(a) and D_1(a) are integral operators defined by f(a)(()(q) = J-/dpe i P c l / ' ] f(p)Jdq'e- i P c l' / J (t)(q'). 2nJ

(15.17)

Thus, the continuous base states (ql defined in (15.8) satisfy the minimum requirement for a coherent state, i.e., that (ql is a strongly continuous function of q that satisfies the relation (15.15). Nevertheless, the most important physical p r o p e r t y of (ql is given by (15.16), (15.13), (15.7), namely that the position operator Q. has neither eigenstates nor eigenvalues.

Instead, it has only a fuzzy

value with an uncertainty Aq a R. In other words, a "point particle" (i.e., one with no size and no structure), if measured, will never be found at one and only one point q at a time. Thus, it is appropriate to term (ql and Q_fuzzy states and fuzzy dynamical

variables,

respectively.

(In principle, both P and Q.should be

fuzzy dynamical variables if the length scale S is treated as a finite quantity.) Formally, (ql is an eigenstate of the operator

aP with the eigenvalue q.

However, such an operator Q.rj has no operational

meaning in the present formalism of fuzzy quantum mechanics. The fuzziness of the base state (ql is completely determined by the function D(p). Let us consider a plausibility argument for the function D(p) to have the following form:

218

A Broader View of Relativity

2

a4 =

D(p) = 2

2

a p +l

2RZ

(15.18)

r

Classically, one can determine a particle's position by confining the particle in a certain range Ax by an attractive square-well potential. If the potential becomes narrower and deeper, we can determine the position more accurately. In the limiting case, we have a 5-function

potential, -V 0 o(x-x 0 ), and the

classical particle is precisely located at the point x 0 .

However, a quantum

particle is described by the Schrodinger (or Klein-Gordon) equation

.Ji_di2

V06(x - x 0 ) <(>(x) = EKx),

(15.19)

2m dx

rather than the classical Newtonian equation.

The solution to (15.19) has the

form I x - x0 I

<|>(x) = A exp

(15.20)

V2R

V2mV 0 '

which shows that the position of the particle is fuzzy (with the u n c e r t a i n t y Ax~R).

The Fourier transform of (15.20) leads to

f Aexp

IX-

XQ

T/2R

I

exp

i(x-xp)p

J

dx =

2T/2AR 2

p R2/J2+l'

(15.21)

which has the same form as that in (15.18). Thus, it is reasonable to use D(p) i n (15.18) to describe the fuzzy base state (ql in (15.8) for a quantum particle. appears that the value R can only be determined by future experiments.

It

There

is, as yet, no compelling reason to identify R with the Planck length ~10~ 3 3 cm or any other known length in physics. According to the "probability axiom" of quantum mechanics, the physical base states associated with the dynamical variables 0_ and P must have a probabilistic meaning. Thus, these states should satisfy Klauder's postulates of continuous representation rather than the usual eigenbras and eigenkets.

One

might think that this is nothing new because one can always transform the

Common Relativity and Quantum Mechanics

219

fuzzy state (ql in (15.8) to the eigenstate
transformation.

However, this is not true: Under a nonunitary transformation that preserves PQ.-QP=-iJ, the position operator Q.and the states (ql and Iq) become Q. - Q.' = DfoJQD-1 (p) = Q, - ij dln " ( P ) , aP

(15.22)

(ql-fql=(qlD- 1 (P) =
(15.23)

l q ) - l q } = D(P)lqMq).

Thus we see that the expectation value of the coordinate Q. remains the same (qilQJq2)HqilQ:iq2H, i.e., the fuzzy

feature

of

(15.24)

coordinates is

unchanged

by

a

nonunitary

transformation. We postulate fuzziness as a fundamental and inherent property of a particle's position.

Regardless of how

one improves the

experimental

techniques and the apparatus, there will always be an uncertainty AqaR associated with each measurement, even if the particle itself has no size and structure. From this emerges a strange new picture of the fuzzy point particle: Namely, a particle is, at a given instant of time, partially located at one point and partially elsewhere, and can never be completely

at one point.

This is analogous

to Zadeh's original idea of fuzzy set t h e o r y 7 and in sharp contrast to both the classical and the conventional quantum-mechanical concept of a point particle. 15c.

Fuzzy point and modified Coulomb potential at short distances The Klauder representation in Hilbert space naturally allows a basic

length scale R to characterize continuous states.

As long as R>0, our results

indicate that space is continuous but fuzzy at short distances. The length scale R characterizes the smallest width of a wave packet that can be physically realized and the fuzzy base vectors (ql form a submanifold in Hilbert space. Since the radical length R seems to be so small that its effects are yet undetected, one might ask what types of physical effects we might expect from this fuzziness and how they might be detected experimentally.

220

A Broader View of Relativity

The fuzziness of

an electron's coordinates implies that the electron must have an " R - i n h e r e n t charge distribution" given by

PRM =

^ - fD 2 (p)expf-ip L (2*J)31

10-\/2R:

r/j]d3p J

r

-exp

(15.25)

V2R 2R/

D(p) =

J2

aV+1

where the form of D(p) is assumed to be given by (15.18). Note that PROD^-e^V)' as R approaches 0, as expected. This R-inherent charge distribution is consistent with the fuzzy point picture of a particle. The modified Coulomb potential VR(T) produced by such a fuzzy point electron is given by

JdV 4JI PIRr(r'I- r' i.e.,

_ f d 3 k D 2 (k) VR(r) = - e5 j —^exp(-ikr), / (2K)3 k2

-e 1 - 1

4«r

k=p/J,

exp[-r/(V2R)] 2T/2R

e

(15.26)

r»R,

4jtr

r«R, 8^2:j t R

which is finite at r=0.

This suggests that the electromagnetic interaction will be

asymptotically free at very high energies (or p »h/K).

Common Relativity and Quantum Mechanics

The results (15.26) and

221

(15.25) indicate that R should make its presence known in say, e + e~ scattering at very high energies e~e+ -> e_e+

or

e _ e + -» \T[i+.

(15.27)

If one measures the differential cross sections carefully, a deviation from the usual quantum electrodynamics results could be attributed to the radical length R, provided that the electron is not a composite particle.

The details of the

deviation are related to the specific form of the function D(p). Of course, there is also the possibility that the constant R in Klauder's continuous representation is so small that it will forever elude detection. In this case, the present formalism

is still advantageous to o r d i n a r y

mechanics because this formalism enables us to apply the

quantum

probabilistic

interpretation to all states of all observables regardless of whether they are discrete or continuous. It is likely that Klauder's continuous representations for coordinates and momenta are not merely a matter of mathematical purism, but r a t h e r

an

inherent property of the physical world at very small and very large distances. Such representations are particularly interesting because they suggest a connection between the microscopic physical world and the new mathematics related to fuzzy set theory. Also, introducing fuzziness at short distances might lead to a solution to the divergence problem in field theories. Note that it is highly nontrivial to accommodate the fuzziness of the coordinates and the usual 4-dimensional symmetry of spacetime in special relativity. If the fuzziness of 0_ is really fundamental, then it would imply that the usual 4-dimensional symmetry of special relativity becomes exact only at low energies or in the limit R->0. 15d. Suppression of the contribution of large momentum states to physical

processes

Let us assume that the inherent fuzziness of the coordinate variables (or of a quantum particle's position) is a fundamental property of physics and is independent of the 4-dimensional symmetry of spacetime. We would expect that, apart from the usual relativistic time dilation (or the dilation of the decay length), the lifetime of an unstable particle decay in flight with a momentum p

222

A Broader View of Relativity

will be further dilated due to the existence of the radical length R when p a J/R. (This new effect may be called the 'radical dilation.') The new 'radical dilation' effect for the lifetime of a moving particle is related to the radical length R. The fuzzy coordinate base state (15.8) leads to 00

3

(q'lq) = (2xJ)- jD 2 (p)e i ( < l'- < l ) P / Jd 3 p = D 2 (a , )5 3 (q'-q),

(15.28)

D2(d')f (q') = (2jtJ)-3 / d 3 p e i p q / J D 2 ( p ) / d ^ ' P ^ f (q),

(15.29)

where

D 2 (p).

[2R2p2/J2+l]2

This property, which will be discussed in section 16c, can be

properly

implemented only in quantum field theory. However, we can still see its partial effect on a physical process through an effective density of states, without invoking quantized fields. We first note that mathematically, the momentum p can take any value between -°° and +°°.

However, the region in which the

momentum of a particle can be physically realized is effectively finite.

Thus,

the three-dimensional momentum space appears to be non-Euclidean because one may picture the momentum space as having a volume element E>2(p)d3p, as shown in (15.28).

As a result, the number of one-particle states with a

momentum between p and p+dp is given by the following "effective density of states" [ f d 3 q ] D 2 (P)d 3 P . L J

J

(2JIJ)

3

for a spin-zero particle.

[V]D

2

(P)d3P)

(1530)

(2KJ) 3

Thus the contribution of large momentum states to

scattering cross sections or decay rates of physical processes is suppressed due to the new restriction (15.7) on the Heisenberg uncertainty relation.

Common Relativity and Quantum Mechanics

223

References 1.

J. P. Hsu, Nuovo Cimento B 78, 85 (1983); J. P. Hsu and S. Y. Pei, Phys. Rev. A

2.

W. Heisenberg, Ann, d. Phys. 32, 20 (1938); P. A. M. Dirac, Rev. Modern

3.

R. P. Feynman, Quantum

4.

J. Schwinger, Quantum

5.

R. P. Feynman, Theory

6.

E. Wigner, private conversations ( Univ. of Texas at Austin in April, 1976).

37, 1406 (1988). Phys. 21, 392 (1949), see p. 399. Electrodynamics

(Benjamin, New York, 1962) pp.

138-139. Electrodynamics

(Dover Publications, New York,

of Fundamental

Processes

1958) p. xvi . (Benjamin, New York,

1962) p. 145. For a related discussion, see S. Weinberg, The Quantum Theory of Fields, vol. 1, Foundations (Cambridge Univ. Press, Cambridge, 1995) pp. 31-38. 7.

L. A. Zadeh, Inf. Control 8, 338 (1965); L. A. Zadeh, K. S. Fu, K. Tanaka and M. Shimuras, Fuzzy Sets and Their Applications

to Cognitive

and

Decision

Process (Academic, New York, 1975). 8.

J. Klauder, J. Math. Phys. 4, 1055 (1963) and 4, 1048 (1963); J. P. Hsu and S. Y. Pei, Phys. Rev. A 37, 1406 (1988); J. P. Hsu and Chagarn Whan, Phys. Rev. A38, 2248 (1988).

224

A Broader View of Relativity

16. Common Relativity and Fuzzy Quantum Field Theory 16a.

Fuzzy quantum field theories It has been observed that finite quantum electrodynamics cannot be

formulated consistently within the framework of special relativity. 1 The reason for this is that limiting the magnitude of interactions while retaining

the

customary coordinate description is contradictory, since no mechanism is provided for precisely localized measurements. This observation suggests that the physical properties of space at very short distances may be very different from those at macroscopic or even atomic distances. The fact that no mechanism is provided for precisely localized measurements indicates that there might be an inherent fuzziness at short distances and that the concept of a point-like particle, which works very well in the atomic and nuclear domains, must be modified so that it resembles some sort of n o n - p o i n t - l i k e object.

Spatially

extended particles, fuzzy particles and s t r i n g - l i k e objects have

all been

discussed in the literature.

2

However, it is very difficult to make extended

particles and string-like objects consistent with the 4-dimensional symmetry of the Lorentz and the Poincare groups. Thus far, people have found strings that fit a symmetry framework of a 26-dimensional or 10-dimensional spacetime, but this has as yet little relevance to present experiments and observations.

Rather

than trying to explain why a 10- or 26-dimensional spacetime has only 4 observable dimensions, a different approach would be to find a string-like object (no matter how strange) that can be used to formulate a consistent theory in the 4-dimensional spacetime.

If this can be accomplished, it would be an

important discovery. On the other hand, one may ask:

Is the 4-dimensional symmetry of

Lorentz and Poincare invariance so sacred that it must be absolutely preserved at all costs? A physical principle is sacred only as long as it is supported by experiments. However, experiments have limitations: Newton's law of motion is known to be good for the macroscopic world and small speeds, and the Dirac equation is good for spatial regions where Ar>10~17 cm and for energies smaller

225

than roughly 10 3 GeV. Nevertheless, whether or not these physical principles are valid for describing the early universe or for particles when their e n e r g y reaches 10 3 0 GeV is far from certain. Based on the evolution of physical theories and discoveries of physical laws in the past 100 years, it seems reasonable to conjecture that most currently established physical laws will not last another 100 years without modification. Dirac commented on the divergence difficulties in the quantum theory of fields: "The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr's orbit theory to the present

quantum

mechanics." 3 It is reasonable to assume that the 4-dimensional symmetry of Lorentz and Poincare groups is a good starting principle, which can be realized i n different conceptual frameworks, as discussed in previous chapters.

We now

explore the physical origin and implications of the concept of fuzzy particles, which can be accommodated in the 4 - d i m e n s i o n a l

symmetry of common

relativity (but not that of special relativity). It appears quite possible that the divergence difficulties originate from the naive analogy between field

quantities

(cp(t,r), dq>(t,r)/dt) and

the

generalized coordinates and velocities (qi(t), dq;(t)/dt). For decades, it seemed quite certain that one had to follow this "exact analogy" in order to apply the canonical procedure for quantizing the fields.

However, if one insists on

preserving this "exact analogy," one is led to the customary coordinate description at short distances and thus encounters profound

divergence

difficulties. This leads to some doubts about to what lengths this analogy can be extended.

According to this analogy, one simply replaces the

coefficients

c k and c k of a free

classical field

q> by the

Fourier

corresponding

annihilation and creation operators:

at

ck-*-5k-, ^/2(O k

c*k -*-rJS— -y/2(0k

In this way, the usual Hamiltonian H u s u of the field q> is

226

A Broader View of Relativity

(16.D

HUsu = 2 "k a k a + + - , k

*

(162)

'

where the field qp is enclosed in a cubic box of side L=V1/3 and satisfies the periodic boundary conditions. Result (16.2) shows the analogy with the harmonic oscillator. In this section, we attempt to modify the conventional quantum field which allows, in principle, the existence of a physical wave with an infinitesimal wavelength, so that one can measure spatial positions to an arbitrary accuracy. The basic new idea is that such a wave does not exist even in principle. Thus, spatial measurements cannot be made as accurate as one wants, even in principle. In this sense, space is "fuzzy at short distances." Such a fuzziness suggests that there exists a fundamental probability distribution P(k,m) that approaches zero as the wavelength or 2rc/k approaches zero. We assume that this is inherent in each harmonic oscillator with the momentum k of a field with a mass m. 4 Operationally, this means that instead of (16.1), one should replace the Fourier coefficients of a free classical field by

c

_ akD(k,m) V2cok (16.3)

c»k

ajD(k,m) -* k . V2cok

D(k,m) = Vp(k. m >.

k = Ikl.

Thus, the Hamiltonian of the field q> becomes

H = 2) o) k P(k,m)a k aj c + i.

(1 6 . 4 )

The specific form of the probability distribution P(k,m) can only be determined by future experiments. Nevertheless, based on known experiments at low energies, the probability distribution P(k,m) must satisfy the following limits:

Common Relativity and Fuzzy Quantum Field Theory

227

1,

k - 0,

0,

k ^ oo.

P(k,m)-

(16.5)

As we saw earlier in chapter 15, the concept of an i n h e r e n t

probability

distribution has the effect of suppressing the large momentum contributions, this time to the harmonic oscillator Hamiltonian (16.4).

In addition, field

oscillators would have a finite zero-point energy, in contrast to the usual expression (16.2) which has an infinite zero-point energy.

To make our

discussion concrete, we will assume the invariant function D(k,m) in common relativity to be given by (15.3) with p=Jk,

G(k) = ^ k 2 + r n 2 ,

P(k,m):

2

(16_6)

2

l + I G (k)

where the quantity I is a new fundamental constant. Presumably, the value of I is very small so that P(k,m)=l for presently

attainable energies

in

the

laboratory. We may picture a field as a set of harmonic oscillators where each harmonic oscillator is associated with an intrinsic probability determined by its momentum. In the limit I-»0, the probability distribution P(k,m)-»1 and we h a v e the usual field in which all states of the harmonic oscillator of the field are equally probable. The probability distribution P(k,m) in (16.5) must have the following two important properties: (i)

P(k,m)

depends

only

on

four-momentum vector of an oscillator.

the

spatial

components

of

the

This dependence is necessary for the

unitarity of quantum field theories, i.e., the probability for any physical process to occur must be non-negative. (ii)

P(k,m) must be an invariant function,

as required by

the

four-dimensional symmetry. These two requirements can only be satisfied by field theories formulated within the framework of common relativity because special relativity does not have the invariant

genergy (15.2), which is related only to the

spatial

components of the momentum four-vector. Thus, quantum field theories based

228

A Broader View of Relativity

on special relativity cannot satisfy (16.5) or possess the two properties described above. Furthermore, the experimental evidence for the presence of P(k,m)*l in the future would imply that the spacetime symmetry of special relativity is only approximate at "low energies" and is not exact at very high energies or short distances. According to Maxwell's equations and the postulate of the universal probability distribution for field oscillators, we expand the photon field A(oo,r) as follows,

A(ro,r)= .

l

f

V(2J0 3

d3k

^e(k,X)^l?(k,0) [a(k,Me- ikx + a t ( k , M e i k x ] , (16.7) L

J V2k7 x-i

k0 = Ikl;

in the radiation gauge.

k-e(k,>,)=0;

e(kA)-e(k,>0 = 8xv ,

J

k2 = k ^ = 0,

Since photons are massless, they are associated with t h e

invariant probability distribution P(k,0) in (16.7). The operators a ( k > ) and at(k,X) are assumed to satisfy the commutation relations,

[a(k,^),at(k',V)] = 5 3 (k-k')5^, (16.8) f

f

[a(k, X), a(k', X')] = [a (k, X), a (k', X')] = 0.

As usual, we assume the gauge invariant Lagrangian L a for the photon field a^:

L a =-—f[j,vf|iv;

f^v

=a

n a v ~dva\i-

(16.9)

Note that a?- =(a°,a) in common relativity has the same dimensions as A>Vc in special relativity because L a d 4 x and J have the same dimensions.

The

Hamiltonian H a and the momentum Q. a of the photon field can be written as

Common Relativity and Fuzzy Quantum Field Theory

229

Ha = J d3k I k I PCk.O)f a G k . ^ C k , * . ) ,

(16.10)

Q.a = J d3kkPCk, 0) I aCk, X)afCk, X). J x=l The modified photon propagator is now given by

iD^(x,I) -<0la Ii (x)a v (0)8(w)l0>+ <0lav(0)a(i(x)e(-w)l0>

/

d 4 k P(k,0)e-' k x | E ( i ( k x ) E v ( k A ) ) (In) k +ie x=l

(1611)

E^Ck.x.) = fo,£Ck,l),8Ck,2),k/|k|) =(0, E(k, X)).

As expected, this propagator reduces to the Feynman propagator iDprv(x) in the limit 1^0,

iD»( X ) 0) = iD^ v (x).

(16.12)

The Coulomb potential of a charged particle is now produced by a fuzzy source rather than a point source. probability function P(k,0) in

The fuzzy source is determined by the

(16.11). 5

The modified Coulomb potential

V(r)=a 0 (r) is

V(r) = _e I A f M e - 4 ' H , (2ny k'

J a

(16.13)

which is consistent with (14.26). It is important to remember that P(k,0) should not be understood or interpreted as the form factor due to charge distribution i n the classical sense. Rather, it is a quantum mechanical property related to the inherent probability distribution of photon field oscillators.

230

A Broader View of Relativity

In other words,

photons should be pictured as a "fuzzy particle" rather than a point particle. A "fuzzy photon" can produce an effect similar to that produced by a non-point charge distribution.

In general, any departure from the point picture of

particles in electrodynamics implies a modification of the Coulomb potential at short distances which can be tested by very high-energy experiments in the future. 16b. Fuzzy quantum electrodynamics based on common

relativity

In fuzzy quantum electrodynamics (fuzzy QED), the invariant action Sq, involving a fuzzy electron field i|> and a photon field a^, is assumed to have the usual form

Sq = J"Ld4x,

L=Tj) ^ ( U a ^ - e a J - m

J = 3.5177xlCT 38 g-cm,

i-iv1".

(16.14)

e = -1.602xlCr 20 V4jt ( g c m ) 1 / 2 .

(16.15)

The wave equation of a free Dirac field can be derived from (16.14). It has the usual form

Y^ija^-m x|) = 0.

(16.16)

Fuzzy quantum electrodynamics is based on the general postulate that there is an inherent probability distribution P(p,m) associated with a field oscillator with momentum p and mass m. This is a natural generalization of the idea of the "fuzzy quantum field" discussed in section 15 a. Based on the 4-dimensional formalism with a covariant gauge condition, we have the following free photon and electron fields:

i x/ a (l (w,r) = 2 , ^ M [ a ( p , a ) E | i ( a ) e - P ' J + at(p, a ) E (aJe'P'^Jl

5,'aV

2V

P0

L

Common Relativity and Fuzzy Quantum Field Theory

J

(16.17)

231

if(w,r)=yJ^E^[b(p,s)u(p,s)e-iPx/J + d t (p,s)v(p,s)e i P x / J l p;sV V P0 L J

(16.18)

where p x = p x1*,

aCp.aJ.a^p'.a') = 8 pp '6 aa <, (16.19) [ t y p . s ^ b V . s ' ) ] = 8PP'6SS.,

[dfosKd^p'.s')] = 8 pp .8 ss .,

and all other commutators vanish. Of course, commutators for quantized fields \|>(w,r) and a (i (w,r) can be derived from (16.17)-(16.19). In a general inertial frame, the Dirac equation in (16.16) can be written as

U ^ - = [-iJa D -V + p D m ] ^ . aw

(16.20)

As before, w = bt c denotes the lightime, t c is common time, and a D , pD are t h e usual constant Dirac matrices. Since the lightime w plays the role of evolution variable in the invariant equations of motion (16.20), it is also the evolution variable for a state o (s, (w) i n the Schrodinger representation

.Jao^(w)=H(s)(w)(I)(s)(w)) aw

H^H^+Hf), '

(16.21)

because the evolution of a physical system is assumed to be described by a Hamiltonian operator which has t h e same transformation property as t h e lightime w or a/aw. The conventional covariant formalism of perturbation theory can also be applied to fuzzy QED.

For simplicity, we set J=l, so that the dimensions of t h e

Lagrangian density L, the photon fields a„ and the electron field xp are

232

A Broader View of Relativity

[L 1 / 4 | = [a^] = L 2 / 3 l = [mass] = [l/length],

J = l.

(16.22)

To obtain the modified rules for Feynman diagrams in fuzzy QED, we follow the usual quantization procedure and define the fuzzy QED Lagrangian LFQED by adding a gauge fixing term in the Lagrangian (16.14),

LFQED = L - ^ ( a % ) 2 ,

J = l,

(16.23)

where p is a gauge parameter. We define the M-matrix as follows: Stf =8 lf -i(2*)464(p -

P

|

t o t )

) J w ( y V ) Mif,

(16.24)

where "ext..par" stands for "external particles." The quantity nj denotes mj/p 0 j for spin 1/2 fermions and l/(2poj) for bosons.

Because of the 4-dimensional

symmetry in (16.23) and (16.24), the rules for writing Mjf are formally the same as those in the usual QED: (a) The fuzzy photon propagator is now given by -iP(k,0)h v -(l-p)k kv / k 2 ] — ^ >* -, (k 2 + ie)

k2=k„k^. "

(16.25)

(b) The fuzzy electron propagator is

iP( P'm) . Y^-m+iE

(16.26)

(c) The electron-photon vertex is

-(ieV-

(16.27)

Common Relativity and Fuzzy Quantum Field Theory

233

(d) Each external photon line has an additional factor VP(k,0) e^. Each external electron line has VP(p,m)u(s,p) for the absorption of an

electron

and

VP(p,m) u(s,p) for the emission of an electron, etc. Other rules such as taking the trace with a factor (-1) for each closed electron loop, integration with d 4 k/(2jt) 4 over a momentum k^ not fixed by the conservation of four-momentum at each vertex, etc., are the same as the usual Feynman rules. Thus, calculating scattering cross sections and decay rates (with respect to the lightime w) of a physical process, one obtains formally the same result as in conventional QED.

For example, let us consider again the decay rate

r(l-»2+3+...+N) for a physical process 1^2+3+...+N given by

r(l-2+3+...+N)=lim

fKflSli)l2d3x2d3p2^d3xNd3pN|

w—J

w

( 2 j t j)

3

(1628)

(2TCJ)3

where w=bt c and r(l-»2+3+...+N) has the dimensions of inverse length.

Its

inverse is a particle's lifetime measured in terms of the lightime w=bt c w h i c h has the dimension of length and hence, 1/r may be called the "decay length." The decay length D is given by

D=

. r ( 1 ^ 2 + 3 + ... + N)

(16.29)

Thus in common relativity, one has the "rest decay length" T\, for the decay of a particle at rest, corresponding to the "rest lifetime" in the conventional theory. Also, instead of the dilation of the lifetime of a particle in flight, we have a dilation of the decay length it travels before decaying. This result is consistent with experiments because it is equal to the experimentally measured distance traveled by a decaying particle. (See chapter 10, sections lOg and lOh.) For a scattering process 1+2 -»3+4+....+N, the differential cross section da, which has the dimension of area, is given by

234

A Broader View of Relativity

IM if l 2

dO:

4[(Pi'P2) 2 -(m 1 m 2 ) 2 l

d3pN (2ji) 3 2p 0 N

liefer) ext.fer

d3p3 (2^) i 2p 0 3

(2nr54(p1+p2-P3-P4-

(16.30)

PN)SO,

where po = (p 2 +m 2 ) 1 / 2 and S 0 denotes a factor 1/n! for each kind of (n) identical particles in the final state. If the initial particles are unpolarized, one must take the average over initial spin states. process, then [n e xt fer (2mf er )] The formal

expressions

When there is no external fermion in a

in (16.30) is replaced by 1. (16.28) and (16.30) are

the

same as

the

conventional ones. The new and different effects in fuzzy QED come from the M-matrix, which involves modified propagators related to an inherent fuzziness at short distances as shown in eqs. (16.25) through (16.27). 16c.

Experimental tests of the 4-dimensional

symmetry of

special

relativity at very high energies Let us elaborate what new physical effects may be obtained due to an inherent probability distribution P(k,m) for field oscillators.

Suppose one

calculates the differential cross-section of unpolarized electrons scattered by an external potential (or a point-like nucleus). One obtains the result

do dQ

-(Za) 2 P( Pi -p f ,0)P( P i ,m)P(p f ,m)

(jPi I4 sin4(8/2)J

x(l-p2sin2(e/2)\

P(Pi-Pf,0)-

l/tPi-Pf) 2

(16.31)

P(Pi,m):

*?

2 +

Q

m2

_ _£. = C,'

Common Relativity and Fuzzy Quantum Field Theory

(16.32)

235

P(p f ,m) ;

.*F

IPi I

IPi-Pf l 2 =4lpi l2 sin^2 9

(16.33)

cf If the laboratory is chosen to be the F frame in which the speed of light is defined to be isotropic, then Qi=Q=Cf=c=29979245800cm/s (see section 12c). The differential cross section for the scattering of unpolarized electrons (16.31) is suppressed at large momenta by the inherent probability distribution associated with the photon in the intermediate state and the electrons in external states. In this calculation, the point-like nucleus is assumed to be at rest and its inherent probability distribution is negligible. Next, let us consider the differential cross section for Moller scattering, (after C. Moller who first discussed the process in 1931). -

Nteller scattering is -

the scattering of two electrons, e~(pi)+e (p 2 )—e-(pi)+e (p 2 ).

Since we are

interested in the effects of the inherent probability distribution, we calculate the differential cross section at "high energies," p 1 0 » 0 . The result is relatively simple to obtain in the center-of-mass frame,

d o = i £ jn_ dQ 8 ^pioj

P

4 4 t a m ) l + cos (e/2)p2 sinW^

|

l + sin4(8/2)p2 cos^ J/2)

2P_P\ cos 2 (8/2)sin 2 (9/2)

P = Pi = -P2.

P' = Pi = -P'2>

P ± =P(k ± ,0),

k ± =^j±p'[

(16.34)

-H-K r 0 = ^ = 2.82xl(r i ; ( cm, m

where one may choose the center-of-mass frame to be the frame F in which the speed of light is isotropic. One may also consider the pair annihilation process, i.e., e~(p_) + e + (p+) -» Y(ki) + v(k 2 ). In the laboratory frame, in which p_=(m,0,0,0), the differential cross section is

236

A Broader View of Relativity

do _ r ok 1 0 1 P(p_,m)P( P+ ,m)P(k 1> 0)P(k 2 ,0) do 8 IJp+l(m+p+0)

k10 k2o

2(h.k2Y k 2 om

2_4(ei.E2)2 +

k 20

^.k^

k

k

10

10m

2^l)i+2(£Lk2)i PaPb k10m k20m

Pa = P(P_ - ki, m ) ,

Pb = P(p_ - k 2 , m ) ,

(16.35)

kf = (k i 0 kO,

i = 1,2.

In the limit of zero radical length, R^-0, (i.e., I^>0 in (16.6)), all i n h e r e n t probability distributions reduce to 1, and the terms involving ^ - k ^ and (ei-k2) cancel, so that (16.35) reduces to the usual result. The new properties of the scattering cross sections (16.31) to (16.35) due to fuzziness at short distances can all be tested by future

high-energy

experiments. These are also experimental tests of the possible inexactness of the 4-dimensional symmetry as instantiated by special relativity at very

high

energies and short distances. At present, there exists experimental data for the decay of unstable particles in flight

at high energies (several h u n d r e d s

of GeV), K s (p)

-»JT+(P+) + nr(p_). This can be used to estimate the upper limit of the radical length R in this theory. The decay rate of Ks in flight is roughly modified by the inherent probability distribution as follows

r(K° -* JI+ + JT) . Po

P(p, m K )P(p + , m ^ )P(p_, mK-) x const.

(16.36)

At several hundred GeV, the masses of the mesons can be neglected in the estimate. The departure of P(p,mK)P(p+,mIt+)P(p_,mJ1._) from 1 must be within the experimental error. Based on the result of the decay rate of Ks

in flight, 6 we

estimate that

Common Relativity and Fuzzy Quantum Field Theory

237

I<2xl(r4— mp

or

R<4xlCT13cm,

(16.37)

where we have ignored the meson masses. Since mp=lGeV/c2, (16.37) and (16.6) indicate that the inherent probability distribution P(k,m) becomes important only at extremely large momenta ( p » 1 0 4 GeV/c2). In summary, if we treat the concept of probability in quantum mechanics as fundamental so that the realizable position states of a particle must have a probabilistic interpretation, then the contributions of the corresponding states with large momenta will be suppressed by an inherent probability distribution P(p,m). This implies that dilation of decay lengths should come from both the usual relativistic effect and the 'radical dilation' due to R*0 at very high energies. Note that the inherent probability P(p,m) is not a scalar under the Lorentz transformations of special relativity. Therefore, the spacetime 4-dimensional symmetry as expressed by the equations of special relativity is only approximate at extremely large momentum if R*0 or P(p,m)*l. Only in the limit of low energies or R^-0, does the spacetime symmetry of special relativity become exact. However, because the inherent probability P(p,m) can be expressed in terms of the genergy G(p), which is an invariant in common relativity, common relativity provides a framework in which the 4-dimensional symmetry remains exact even in the presence of the inherent probability distribution P(p,m)*l.

238

A Broader View of Relativity

References 1.

J. Schwinger, Quantum

Electrodynamics

(Dover Publications, New York,

1958) p. xvi. 2.

M. Kaku, Quantum Field Theory, A Modern Introduction

(Oxford Univ. Press,

New York, 1993) pp. 699-736; S. S. Schweber, Introduction

to

Relativistic

Quantum Field Theory (Row and Peterson, Evanston, 111. 1961); see also ref.4. 3.

P. A. M Dirac, The Principles of Quantum Mechanics

(4th ed. Oxford Univ.

4.

J. P. Hsu, Nuovo Cimento B 89, 30 (1985); B 78, 85 (1983); J. P. Hsu and S. Y.

Press, 1958) p. 310. Pei, Phys. Rev. A 37, 1406 (1988); J. P. Hsu and Chagarn Whan, Phys. Rev. A, 38, 2248 (1988). 5.

J. P. Hsu and S. Y. Pei, Phys. Rev. A 37, 1406 (1988).

6.

The lifetime or the decay-length dilation predicted by the 4-dimensional symmetry of the Lorentz group has been tested and confirmed

by

measuring the decay lifetime of k°s in flight at several hundreds of GeV (i.e.,Y~103.) See N. Grossman, K. Heller, C. James, et al, Phys. Rev. Lett. 59, 18 (1987).

Common Relativity and Fuzzy Quantum Field Theory

239

17. Extended Relativity: A Weaker Postulate for the Speed of Light 17a.

Four-dimensional

symmetry

as a guiding

principle

Only at the beginning of the twentieth century after the creation of special relativity was it recognized that the concept of symmetry played a fundamental role in physics. 1 The 4-dimensional spacetime symmetry has been one of the most thoroughly tested symmetry principles in physics. In his Nobel Lecture, C. N. Yang made the following piercing observation: "Nature

seems

representations

to

take

advantage

of the symmetry

elegance

and the beautiful

involved

and contrast

laws.

perfection

of

the

simple

mathematical

When one pauses to consider of the mathematical

it with the complex

and far-reaching

the

reasoning physical

consequences,

a deep sense of respect for the power of the symmetry

never fails to

develop."2

laws

This quotation summarizes the essence of symmetry in physics, w h i c h will be illustrated below by an analysis of different viewpoints of the physical world to show how 4-dimensional symmetry is critical to any theory. Let us consider the viewpoint of Reichenbach's more general concept of time and Edwards' universal two-way speed of light, which includes relativistic time and special relativity as a special case. Unfortunately, Reichenbach's and Edwards' treatments were, in general, not consistent with the Lorentz and Poincare symmetry groups and so their efforts never led to an experimentally consistent theory. The lack of the 4-dimensional symmetry of the Lorentz group will cause many problems in the formulation of q u a n t u m field theories, especially the Feynman rules for calculations in quantum electrodynamics or chromodynamics.3

For the purpose of comparison, we shall explain in some

detail the application of 4-dimensional symmetry to the ideas of Reichenbach

240

and Edwards.

In this way, we will also show how their work can be made

consistent with experiments. We term such a 4-dimensional theory "extended relativity."

In

the

context of the family of relativity theories that we have been discussing, including taiji relativity, common relativity, and special relativity, extended relativity is simply taiji relativity with a second postulate stating that the 2-way speed of light is a universal constant. In the following discussions of "extended relativity", involving

the

universal two-way speed of light c, we shall use the usual units for space, time and mass, and retain the usual constants such as e, c and h. Using 4-dimensional symmetry as a guiding principle for discussing physical laws, we first analyze Edwards' original attempt in 1963 to formulate a relativity theory based on a weaker postulate for the speed of light.

Edwards

postulated that "the two-way speed of light in a vacuum as measured in two coordinate systems moving with constant relative velocity is the same constant regardless of any assumptions concerning the one-way speed." 4

He derived

space and time transformations involving infinitely many possible physical times which could be physically realized through Reichenbach's convention of clock synchronization (which includes relativistic time as a special case). Edwards' transformations were shown to be consistent with many experiments related to the propagation of light. Nevertheless, we show that they do not form the Lorentz group so that, in general, physical such

transformations.

expression

for

the

laws are not invariant

As a result, Edwards' work leads to an relativistic

Lagrangian formalism.

energy-momentum

under

incorrect

of a particle in

the

3

In view of these results, one may conclude that assuming the universality of the 2-way speed of light and Reichenbach's concept of time is experimentally untenable. However, this is not the case. In the following paragraphs we show that Reichenbach's general convention of time and Edwards' universal 2-way speed of light can be accommodated in a 4-dimensional formalism which is consistent with the relativistic energy-momentum of a particle and the Lorentz and Poincare groups.

Such a theory, extended relativity, includes special

relativity as a special case. These results are physically and pedagogically interesting following reason:

for

the

Physics students are usually puzzled by discussions of

Extended Relativity

241

Reichenbach's convention of time and the impossibility of the unambiguous measurement of the one-way speed of light in the literature.

They ask: Can

Reichenbach's convention of time be consistent with the Lorentz properties of transformation between two inertial frames?

group

Are the ideas of

Reichenbach and Edwards viable? Our results suggest that the key to answering these questions is the 4-dimensional symmetry of the Lorentz group.

The

physical theory of taiji relativity based solely on the first postulate of relativity, i.e., the principle of relativity for physical laws, has been discussed in chapter 7. We show that the correct formalism of Edwards' theory based on a weaker postulate for the speed of light can be obtained by imposing an additional postulate of the universal 2-way speed of light onto taiji relativity.

Thus,

extended relativity gives a more restricted view of the 4-dimensional physical world than that of taiji relativity.

It also shows the power of the first postulate

of relativity from the vantage point of 4-dimensional symmetry. 17b. Edwards'

transformation

with

Reichenbach's

time

Let us consider two inertial frames F and F', with relative motion along the x/x'-axes.

Suppose there are two identical clocks, clock 1 located at the

origin of the F frame and clock 2 at point x on the x-axis. A light signal starts from the origin

(event 1) at time t x , it reaches clock 2 (event 2) at time t2 and

returns to the origin (event 3) at time t3. Reichenbach's concept of time is realized by synchronizing clock 2 to read t2 by the relation t 2 = ti + E(t3 - t O ,

(17.1)

where e is restricted by 0< E <1, so that causality is preserved, i.e., t2 cannot be earlier than ti.

The same synchronization procedure can be performed on

clocks in the F' frame: t' 2 = t' 1 + E'(t' 3 -t'i).

(17.2)

By definition, the two-way speed C2W of light along the x-axis in F is given by

242

A Broader View of Relativity

c2w=-^-, l

(17.3)

3-li

which is independent of t2 and Reichenbach's parameter e. Similarly, in the F' frame, we also have a constant two-way speed of light,

c'2w = T T ^ r . t3-tl

(17-4)

Following Edwards, we shall assume these two-way speeds of light to have the same value, c

'2w = C2w= c -

(17-5)

The synchronization of clocks in F and F' according to (17.1)-(17.5) defines the Reichenbach time which clearly includes relativistic time as a special case (e = e' = 1/2). In special relativity, an Einstein clock at x is synchronized according to tg = t 0 + x/c (where t 0 is the time of the clock at the origin of F). corresponding Reichenbach clock at x reads tR = t 0 + 2EX/C2 W

S

The

t, which follows

from (17.1), (17.4) and (17.5). Thus, tE and t are related by

t E = t - (2E - 1 ) - = t - 3^, c c

q = 2E - 1.

(17.6)

Similarly, in the F' frame, we have the relation

t'E = t ' - S ^ , c

q'=2E'-l.

(17.7)

For simplicity and without loss of generality, we set q = 0 in (17.6), so that physics in the F frame is identical to that in special relativity.

We now

concentrate on physical implications of (17.7) in the F' frame to see w h e t h e r Edwards' transformation is inconsistent with experimental results. Using (17.6)

Extended Relativity

243

withq = 0, (17.7) and the Lorentz transformation involving Einstein time tE and t'E, one obtains the Edwards transformation between inertial frames F and F',

tf-Ja-pq'tt-fP-q')^!

x'= Y (x-pct),

y'=y

z'=z;

(17.8)

T/TP 1 where V is the speed of F' as seen from F. This type of transformation with Reichenbach's time was first derived and discussed by Edward. 4 It is important to note the following properties: (A)

The coordinate vector (ct',x',y',z') is no longer a 4-vector u n d e r the

transformation (17.8) because (17.8) does not have the explicit 4-dimensional symmetry of the Lorentz group in general. (B)

In the limit V-»0, i.e., F and F' become the same inertial frame, but (17.8)

does not reduce to the identity transformation:

t'= ( t - q ' - l

x'=x,

y'=y

z'=z.

(17.9)

This shows that Edwards' transformations do not form the Lorentz group, except in the special case q'=q=0. (C)

Under Edwards' transformations, one has c 2 t , 2 +2q'x'ct'-x' 2 (l-q' 2 )-y' 2 -z , 2 = c 2 t 2 - x 2 - y 2 - z 2 ,

Note that if q * 0 , t h e n x2(l-q2)-y2-z2.

c 2 t 2 -x 2 -y 2 -z 2

will

be

replaced

(17.10) by

c 2 t 2 +2qxct-

It can be shown that this quadratic form holds also for

infinitesimal intervals.

244

q = 0.

A Broader View of Relativity

17c.

Difficulties of Edwards' transformation

The Edwards transformation (17.8) has been shown to be consistent with many experiments related to the propagation of light. 4 Moreover, since Edwards' transformation appears to be obtainable by a "change of time variables," (17.6) and (17.7), in the Lorentz transformation, one might think that it is equivalent to the Lorentz transformation. However, this is not the case because Edwards' transformation (17.8) does not have the Lorentz group properties (except when q=q'=0) due to the transformation of t. 3 Thus, Edwards' transformation (17.8), as it stands, violates the 4-dimensional symmetry of the Lorentz group. As a result, one can show explicitly (see (17.14) below) that the time t' leads to an incorrect expression for the relativistic energy-momentum of a particle in the Lagrange formalism. This contradicts experiments in general, except when q'=q=0. To wit, let us consider Edwards' transformation (17.8), where we have chosen a frame F in which the Reichenbach time equals the Einstein time by setting q=0. We stress that this frame can be chosen arbitrarily. (If one wishes, one can set q'=0 instead of q=0. The following arguments still hold with F and F' interchanged.) For a free particle, we have the actions S and S' in F and F' respectively: S = -J"mc ds = J L dt,

ds2 = c2dt2 -dx 2 -dy 2 -dz 2 ,

L = -mc2Vl-v2/c2, S'= -Jmc ds' = /L'df,

in F (with q = 0),

(17.11)

ds' 2 = c2dt'2 +2q'dx'cdt'-dx'2(1 -q' 2 )-dy' 2 -dz' 2 ,

L'(E) = -mc2T/(1 -q' v'x /c) 2 - v'2 / c 2 ,

in F'(with l>q'>-l).

(17.12)

Note that S = S' because ds = ds' which can be verified by taking differentials of (17.8). The constant c in (17.12) is the universal 2-way speed of light measured in F'. The Lagrangians L and L' lead to the following momenta in F and F' respectively,

Extended Relativity

245

p x (E) = J L flv *

mv

* Vl-(v/c)2

=

p'x(E) = 4 L

,

etc.

2

'

in F,

mcq'-mqV x +mv' x

=

av x

spx

Vd-q'v'x/c) -^'/^

>

etc

(17.13)

^

p,

(1?14)

2

We see clearly that the momentum p' x (E) in F' contradicts experimental results, except when q'=q=0.

Similarly, one can show that the result for the

kinetic energy, defined by (p'(E)-v' - L'), in F' is also incorrect. Before dismissing the expression (17.14), one should check

whether

(17.14) is the same as the result of changing the time variable in the normal special relativistic momentum p' x (SR),

p' x (SR)= ,

mU x

'

,

u ' x - - £ £ - , etc.

Vl-u' 2 /c 2

dt

in F .

(17.15)

E

Under a change of time variable given in (17.7), we have the

relation

dx7dt'=(dx'/dt' E )(dt' E /dt'), etc., i.e.,

u'=-

;*_-. l-q'v'x/c

(17.16)

From equations (17.15) and (17.16), we obtain

Vx p'xCR)- • *p' x (E) , V(l-q'Vx/c)2-v,2/c2

(17.17)

which differs from the momentum p' x (E) in (17.14). This example shows that the lack of 4-dimensional (17.8)

symmetry

in Edwards' formalism

with 1> q' >-1 make it untenable.

based on

transformation

Of course, there are many other

difficulties related to the lack of symmetry such as the non-invariance of the Maxwell equations, the Dirac equation and laws in quantum electrodynamics under the transformations (17.8).

246

A Broader View of Relativity

In the next section, however, we show that if one uses 4-dimensional symmetry of taiji relativity as a guiding principle, all these difficulties can be resolved. 17d.

Extended relativity: Reichenbach's time

A 4-dimensional (a universal

theory

with

2-way speed of

light)

The method for the construction of a 4-dimensional symmetry framework without the usual relativistic time has been discussed before.3.5 The logically simplest case is the 4-dimensional symmetry of taiji relativity which is based solely on the first postulate of relativity. construction of a 4 - d i m e n s i o n a l

It can be applied to guide the

framework

for the present

case

with

Reichenbach's time. For simplicity and without loss of generality, we choose q=0 in synchronizing clocks in the F frame, so that we have the usual relativistic time t = tg. In the F' frame, clocks are synchronized to read the Reichenbach time t'. An event is, as usual, denoted by (w,x,y,z), where w=ct, in F. However, following taiji relativity discussed in chapter 7, the same event must be denoted by (w',x',y',z'), with w'=b't', in F' within the 4-dimensional symmetry framework. We stress that it is necessary (w',r')

transforms

4-dimensional

like

to introduce

a 4-vector

symmetry.

and

the function the

laws

b' so that

of physics

(w',x',y',z')= can

display

Otherwise, the laws of physics cannot be i n v a r i a n t

under the transformation from F to F'. With clocks in F and F' synchronized as discussed previously, the times t and t' are related as in (17.8):

t'=Y (l-pq')t-(p-q')*

c

q'=2£'-l.

This may be considered an assumption.

(17.18)

Indeed, from the viewpoint of taiji

relativity, this is a second postulate which is analogous to assuming the universality of the two-way speed of light over a closed path in any inertial frame, as discussed in section 16b. As usual, we start with the invariance of the 4-dimensional interval s 2 s2=(b't,)2-x,2-y'2-z'2 = c 2 t 2 - x 2 - y 2 - z 2 ,

Extended Relativity

(17.19)

247

to derive the 4-dimensional transformation.

Note that relation (17.19) follows

from the first postulate of relativity, i.e., the Poincare-Einstein 2

2

principle,

2

because it is equivalent to the law of motion, po -p ^n , for a free particle with mass m>0. (See ref. 3 in chapter 6.)

By the usual method for deriving the

Lorentz transformation or the taiji transformation (7.4), we derive the extended 4-dimensional transformation involving Reichenbach's time, w'-b'tWct-px),

x'= v (x-pct).

y'=y

z'=z;

(17.20)

V

Vl-P* where t' is given in (17.18).

This extended transformation

completely

determines the function b'

b'=

^J^ , t(l-pq')-(p-q')x/c

(17.21)

i.e., if x and t are known, we can always calculate the value of b' in (17.20) and (17.21). Note that the fourth dimension

in F' is now b't'=w' rather

than t' or ct',

where w'=b't' is called the "lightime." Although the function b' in (17.21) and the time t' in (17.18) separately have complicated transformation properties, b' and t' are separately well-defined and the product b't'sw' transforms as the fourth component of the coordinate 4-vector.

The properties of b' are

completely determined by (17.21) or the transformation (17.20). From here on, we will use w'sb't' to display explicitly the 4-dimensional symmetry of physical laws. The extended transformation of velocities can be derived from (17.20): , dw' dt'

dt , dt'

„

,

V'X^YWX-PC),

248

(17.22)

4 Broader View of Relativity

dt v vy = — v yv , dt'

, dt vz7 = — vz7 , dt'

where dt/dt' may be obtained from differentiation of (17.18),

1

(dt/df)

dt

' -y ( l - p q ' ) - ( p _ q ' ) ^ L | . dt I cdt

(17.23)

This definition c' * d(w')/dt' in (17.22) is quite natural because dw' 2 - d r ' 2 = 0 can be interpreted as the law of light propagation, c' 2 dt' 2 - d r ' 2 = 0 . Note that the transformation of the ratios v ' / c ' are precisely the same as those in special relativity

v^iv^/cHB. c' l-vxp/c '

ZVc'

y

y/c Y(l-vxp/c)

etc.

(17.24)

Let us consider the property of c' in F'. Its value depends on its direction of propagation. Suppose a light signal propagates along an angle 6 as measured in F and 9' as measured in F'. We have v x = ccose,

v ' x = c'cos 8'.

(17.25)

From the inverse transformation of (17.22), we obtain c = (dt'/dt)[ Y (c'- pv' x )], i.e.,

{(dtf/dtfMi + pcose')]}

c'(8').

(17.26)

Evidently, the average speed of light over any closed path in F is a constant which is equal to c. Now we shall show that the average speed of a light signal over any closed path is also c, even though the speed of light is no longer isotropic in F'. Suppose a light signal travels along the vectors r ' j , where i = 1,2,...N, which form a closed path on the x'-y' plane in F'. The total distance L' to t and the total time T[0t are given by

Extended Relativity

249

N

N

L W - 2r'i .

and

r ..

T' t o t = 2 - *^ 1 -

; N>1.

c

i=l

(17.27)

i = l t© i J

The average speed c'av of the light signal over this closed path is

c'=iitQL T' t o t

Ltot

=

(17.28)

=c

N

L ' t o t / c + (q'/c)2r',cose'i i=l where we have used v' xi = c'(9'i)cos8'i,

(17.29)

c'(e'i) =

(17.30)

dt'| dtjj

r-^ T, (dt'/dt)i[Y(l + pcose'i)] 1 + q'cose'i Y(l + Pcos8'i)'

i = 1,2,....,N,

(17.31)

N 2r'icose'i = 0 .

(17.32)

i=l

Equation (17.32) is a property of a closed path in F'. Thus, the average speed of light over an arbitrary closed path is a universal constant c in extended relativity. We note that a closed path for a light signal as observed by F observers is, in general, not a closed path as observed by F' observers.

Suppose a signal

1

starting from 0 ( o r 0'), i.e., the origin of F (F ), travels to a point A (A') on the x-axis and is reflected back to B = O (B'* O) as observed in F (F'). The two-way speed of this signal in F is c. However, one may ask, what is the average speed of this light signal as measured in F7

From Reichenbach's time (17.18) and

extended transformation (17.20), we derive the space and time coordinates of these events in F',

250

A Broader View of Relativity

t'(O') = x'(O') = 0,

(17.33)

t'(A') = Y [l-Pq')t(A)-(p + q ' ) ^

,

t'(B') = Y [l-Pq')t(B)-(p + q ' ) ^

,

x'(A') = Y((x(A)-pct(A)),

x'(B') = Y((x(B)-pct(B)).

(17.34)

(17.35)

Since t(O)=x(O)=x(B)=0, x(A)=ct(A)=L and t(B)=2L/c, the equations in (17.33) -(17.35) lead to the "average speed" c' av (nc) for such a non-closed path in F':

v

x^A^x^zx^)i tXA'HItXB'J-tHA')!

=

_ ^ _ 1-pq'

Result (17.36) can also be derived from (17.22) directly because this average speed of light can be considered as two events which satisfy

Ax = x(B) -

x(O) = 0. As far as constant velocities are concerned, Ax = 0 is equivalent to dx = 0 or dx/dt = v x = 0. Using relation (17.23) with the condition v x = 0, we obtain

^-Y(l-pq'), dt

vx=0.

(17.37)

Thus using the expression for c' in (17.22) for a non-closed path in F', we also have the relation

C '(v x =0)

=—-—, d-pq')

(17.38)

which is precisely the result (17.36). 17e.

The two basic postulates of extended

relativity

The physical foundation of the extended transformation (17.20) is based on two postulates:

(1) The invariance of the form of physical laws and (2)

Extended Relativity

251

Reichenbach's time (17.18). The latter may also be regarded as a definition of time much like the definition of common time introduced in chapter 8. As we have discussed previously, relation (17.18) is directly related to clock synchronization.

To be specific, we have used the relation between

Reichenbach's and Einstein's times, together with the Lorentz transformation, to obtain (17.18). One may ask: Is it possible to derive the extended transformations (17.20) and (17.18) without using Einstein's time and the Lorentz transformation as a crutch?

The answer is yes.

Instead of postulating (17.18), one can use its

equivalent postulate: Namely, that the 2-way speed of light over a closed path in any frame is a universal constant and independent of the motion of the light sources. This (second) postulate was first made by Edwards for his formalism. 4 Let us now derive the relation (17.18) based on the two basic postulates of extended relativity: [1].

The principle of relativity for physical laws: The form of a physical law must be invariant under coordinate transformations.

(In other words,

physical laws must display 4-dimensional symmetry.) [2].

The 2-way speed of light over a closed path is a universal constant and is independent of the motion of its source. To derive (17.18) from these two postulates, we first observe that, starting

from (17.19) and following the steps from (7.2) to (7.4), we obtain (17.20) but not (17.18).

Since the F' frame moves along the x-axis, its time t' can only be a

linear function of t and x, t'=Pt+Q_x,

(17.39)

where unknown coefficients P and 0_ are to be determined.

From (17.22) and

(17.39), we have

252

CW-fflcra-P),

£ - c ,

(17.40)

CH-^CCUP),

f - c ,

(17.41)

A Broader View of Relativity

where

( | i ) r P + Qc,

(17.42)

(i) 2 - p " Q F -

(17 43)

-

Postulate [2] implies that

-±— + -±— = J±- = ±±-, c'(+)

c'(-)

c'2w

(17.44)

c

and it follows from equations (17.40)-(17.44) that

P + pcQ. = - -

(17.45)

Y

Without loss of generality, we may express Q, in terms of another parameter q' such that (17.39) is more closely related to the form (17.18) in Edwards' transformation,

Q, = - Y ( P + q , ) . c

(17.46)

Relations (17.39), (17.45) and (17.46) lead to

f = Y (l-pq')t-(p + q ' ) - .

(17.47)

This result is precisely the same as the basic transformation of Reichenbach's time (17.18).

Extended Relativity

253

17f.

Invariant action for a free particle in extended

relativity

Although extended relativity involves a class of different concepts of time, realized by Reichenbach's procedures for clock synchronization based on a universal 2-way speed of light, all physical laws have the 4-dimensional form that are identical to those in special relativity. Let us first demonstrate that extended relativity leads to a correct expression for momentum, in contrast to Edwards' formalism.

The i n v a r i a n t

action for a free particle in F' is

S'=-/mc ds'= fL'df,

c = c' 2w = c 2 w ,

ds' 2 = c'2 dt' 2 -dx' 2 -dy' 2 -dz' 2 ,

(17.48)

c' = — . dt'

Note that c in (17.48) is the two-way speed of light, C2W> which is a universal constant. The Lagrangian L' in the F' frame takes the form

L' = - m c c ' V l - v ' 2 / c ' 2 ,

(17.49)

which leads to the momentum p',

p'= ! n c ( v ' / c ' ) • Vl-v'2/c'2

(17.50)

The "energy" p ' 0 is defined as the zeroth component of the momentum by p,o =

( p ^ )

=

c

2

'

These

form

the

m c _

Vl-v' /c'

(1?_51)

2

momentum

4-vector

p' [1 =(p'o,P') w h i c h

satisfies

the

4-dimensional invariant relation p'2-p'2 = m2c2.

254

(17.52)

A Broader View of Relativity

By Noether's theorem, this is the conserved energy-momentum in extended relativity. We see clearly that the momentum p' and the energy PQ in F are consistent with those measured in high energy experiments. As a matter of fact, the momentum p' x given by (17.50) is precisely the same as the momentum p'x(SR) in (17.15) for the F' frame in special relativity. The reason is that the invariant ds 2 in F for special relativity is the same as ds' 2 given by (17.48), [i.e. ds 2 = c2dt'2 -dx'2 -dy'2 -dz'2 = c'2 dt'2 -dx'2 -dy'2 -dz'2 = ds'2 ], so that p'(SR) = m c ^ = m c d i : = p', ds ds'

u'=^l, dt'E

v'=^. dt'

(17.53)

This example shows that in order for Reichenbach's time to be viable it must be embedded within a 4-dimensional framework. The Klein-Gordon equation takes the form h2dj + m2c2)q> = 0.

(17.54)

It can be shown to be invariant under the extended transformation (17.20) as well because the differential operators 3,1= d/dx^,

and

d'^d/dx'^,

(17.55)

are 4-vectors, where x^=(ct,r), x'(*=(w',r') = (b't',r'). It is interesting to see from (17.54) that, although one has times t and t' in extended relativity, the evolution variable in basic 4-dimensional equation is the lightime w'=b't' rather than Reichenbach's time t' in the F frame. This is dictated by the 4-dimensional symmetry of the Lorentz and Poincare groups and is compatible with the evolution variable of taiji relativity. 17g. Comparison of extended relativity and special relativity Although Edwards' original transformations (17.8) can be obtained by a

Extended Relativity

255

"change" of time variables (17.7) in the Lorentz transformations, this does not imply that the two transformations are completely equivalent and theoretically.

experimentally

The reason is that it is not simply a change of variable within

the framework of special relativity. To be specific, such a change of variable violates the second postulate (i.e., the universal constancy of the speed of light) of special relativity. As a result, after the "change" of time variable in (17.7), one is no longer within the same conceptual framework because, strictly speaking, special relativity is a theory which

possess the

4-dimensional

symmetry with the Einstein time t'E which has the usual time dilation property. However, after the change of time variable in (17.7), one has extended relativity which possess the 4-dimensional symmetry with the Reichenbach time t' given by (17.47) which does not have the usual time dilatation property in general. Nevertheless, we show in section 17j that the experimental result of the lifetime dilation of unstable particles decay in flight can be understood within extended relativity in terms of the dilation of the decay length, just as we did for common relativity. We have shown in (17.12) that Edwards' original transformations lead to an incorrect momentum in the Lagrangian formalism because of its lack of 4-dimensional symmetry, as shown in equations (17.13)-(17.17). This shows the first principle Edwards' formalism

of relativity

for physical

of relativity.

that

laws has not been incorporated

in

In particular, (ct',x',y',z') is not a four-vector.

In sharp contrast, the present formalism of extended relativity based on the new transformations (17.20) between the coordinate four-vectors (w,x,y,z) and with

the

4-dimensional symmetry of the Lorentz and the Poincare groups or

(w',x',y',z'), where w=ct and w'=b't', is

explicitly

consistent

the

Poincare-Einstein principle of relativity. If one compares the space-lightime transformations (17.20) with the Lorentz transformations, one sees that cfE = b'f,

(17.56)

where t'g and t' are respectively Einstein's time and Reichenbach's time. note that relation (17.56) does not imply that extended relativity is an change of time variables within

the framework

of special relativity.

We

ordinary

All that it

implies is that if the fourth component ct'E of a coordinate 4-vector in special

256

A Broader View of Relativity

relativity is replaced by b't', where b' is a function, one obtains a n o t h e r 4-dimensional symmetry framework without the universal constant of the speed of light c. Einstein's

Therefore, (17.56) is simply second postulate

weaker second

the direct consequence

by a universal

two-way

of

replacing

speed of light

(Edwards'

postulate).

If one takes the viewpoint that extended relativity is just a change of time variables given by (17.56), one should be able to obtain all results in extended relativity from the corresponding result in special relativity by the relation (17.56). To wit, the momentum (17.50) indeed can be obtained from usual relativistic momentum (17.15) by a simple change of variable (17.56). However, this is not always the case. For example, a plane wave in F' is described by an invariant function exp[i((o't' E -k'T')]

(17.57)

in special relativity. By a simple change of time variable (17.56), one obtains

exp i | b ' t ' ^ - k ' T '

(17.58)

which leads to an incorrect wave 4-vector (co'/c, k').

The correct wave 4-vector

in F' frame should be

(k'°,k')= - ^ , k ' | .

(17.59)

This is the only wave 4-vector which is consistent with the defined relation v'V = c',

(17.60)

in extended relativity. We note that if quantum mechanics is formulated on the basis of extended relativity, the 4-momentum of a photon should be proportional to the extended wave 4-vector (17.59) rather than (co'/c, k'). As a matter of fact, based on equation (17.57) one can obtain the correct invariant phase, involving (17.60) and the 4-coordinate (w',x',y',z') by c h a n g i n g

Extended Relativity

257

both time and quantities related to time (e.g., the frequency). To do this, one must first know the correct wave 4-vector in extended relativity, i.e., one must be able to formulate 4-dimensional extended relativity independent of special relativity. In this sense, it is gratifying that we are able to formulate a 4-dimensional theory of extended relativity, as shown in section 17e. From previous discussions, we conclude that extended relativity and special relativity are two logically different theories, one that has a universal one-way speed of light and the other that has a universal two-way speed of light (which includes the former as a special case.) Nevertheless, both have the 4-dimensional symmetry of the Lorentz group, so that physical laws have the same forms in all inertial frames. (See section 18c.) As with common relativity, extended relativity is consistent with all known experiment. For any invariant law in special relativity, there is a corresponding law of the same form in extended relativity. This suggests that extended relativity is experimentally equivalent to special relativity. We believe that this is yet another piece of evidence that concepts of time (e.g., Einstein's time or Reichenbach's time) and the resulting speed of light (measured in meter per second) in the 4-dimensional symmetry framework are human conventions rather than an inherent property of the physical world. This is the same conclusion that was already indicated by the results of taiji relativity. 17h. An unpassable limit and a non-constant speed of light From the expressions for the momentum and energy in (17.50) and (17.51) or the transformations of velocities (17.22) under extended relativity, one can see an interesting property of the non-constant speed of light c'. Namely, although c' differs in different directions in F', if one compares speeds of various physical objects in a given direction in F', the speed of light is the maximum speed in the universe. This holds for all values of the parameter q' in (17.22). We stress that this "maximum speed" is a more general property than the speed of light being a universal constant because the latter corresponds to the special case q'=0 in (17.22). It is interesting to note that this property was discussed as a "character" of an entirely new mechanics by Poincare in his vision of a relativity theory in 1904: "The velocity of light is an unpassable limit." (See sec. 5c in chapter 5.) Furthermore, even though the velocities c' and v ' measured using

258

A Broader View of Relativity

Reichenbach's time depend on the parameter q', as shown in (17.22) and (17.23), experimental results turn out to be independent of q'. momentum p ' in (17.50) depends on the ratio

For example, the

v'/c'=(dr'/dt')/(dw')/dt'=dr'/dw'

which is independent of time t' or q'. Thus, the restriction on q' in (17.12) is not necessary from experimental viewpoint. The basic reason for these properties is that, within the 4-dimensional symmetry framework, the inherent evolution variable in the laws of physics is the lightime w' rather than Reichenbach's time t', as already indicated by taiji relativity. (See chapter 7.) 17i.

Lorentz group and the space-lightime

transformations

If an object is at rest in the F frame, i.e., v = (0,0,0), the extended velocity transformations (17.22) lead to

C-—^-C(0), i-pq

•7=^7-Vx(0),

Vy-Vz-0.

(17.61)

i-pq This shows that the speed of the F frame, measured from F', is different from the speed of F' measured from F and, moreover, the difference

depends on the

parameter q'. This might be puzzling because common sense tells us that the speed of F measured from F' should be -V. However, as can be seen from special relativity and quantum mechanics, common sense is often a poor guide in modern physics. Let us consider the ratio v' x (0)/c'(0) in (17.61). This ratio is a constant, independent of dt/dt', and satisfies

p.=

_vM0) c'(0)

(17.62)

Thus, using the ratios of velocities, R' and p, one has the symmetry of velocities between F and F'. This is the same situation as in common relativity.

The reason

for the simplicity shown in (17.62) is simply that the real evolution variable i n extended relativity is the lightime w'=b't' rather than the Reichenbach time t' i n F'. The inverse transformation of (17.20) can be obtained. It takes the form:

Extended Relativity

259

ct = Y(b't'+p'x,) = Y'(b,t,+p,x'), x = 7'(x'+p,b't'),

y = y',

z = z';

Y'=Y-

(17.63)

Let us consider also another frame F" moving with a constant velocity Vi = (Vi,0,0), as measured from F and V'! = (V'i,0,0) as measured in F'. Note that the ratio V\/c' is related to Vi by (17.24): Vj=(ViA>P>. C'

(17>64)

a-pvi/c)

From (17.61), (17.62) and (17.64), we see that instead of V or V, one should use the ratio V/c or V'/c' to characterize the relative motion between inertial frames in extended relativity, as this ratio will always be constant and more importantly, is independent of the parameter q'. The 4-dimensional transformations between F and F" are given by b"t"= viCct - Pix),

x" = YICX - Pict),

y" = y,

z" = z,

(17.65)

where

t"=Yi[d-Piq")ct-(Pi - q " ) x ] i .

(17.66)

From (17.20) and (17.65), we can obtain the transformations between F' and F", b"t"=Y'1(b't'-p'1x'),

x'^Y'itx'-p'ib'f),

y" = y',

where

«-&$>-%•

260

°>-^

A Broader View of Relativity

z" = z',

(17.67)

Y'I - YIY[I - PiP] — i - ^ —

yi—r—,

(17.69)

t"- Y ' 1 [a-P'iq")b'f-(p , i-q")x']i.

(17.70)

This result (17.67), together with other properties such as the existence of an inverse transformation and associativity, demonstrates that the set of 4-dimensional

transformations

forms precisely

the Lorentz group.

extended

This shows

explicitly that the Lorentz group can accommodate a weaker postulate for the speed of light.

The Lorentz group properties are the core of 4-dimensional

symmetry and are experiments 17j.

crucial

for

extended

relativity

and to be applicable to quantum field

to be consistent

with

theories.

Decay rate and "lifetime dilation" of unstable

particles

Fundamental wave equations such as (17.53) with (17.54) show that 4-dimensional symmetry dictates the evolution parameter to be the lightime w rather than t in a general inertial frame.

If we consider the decay rate

r ( l - » 2+3+...+N) for a physical process l-» 2+3+...+N, we can follow the steps i n Appendix C because we have 4-dimensional symmetry in extended relativity. This cannot be done in Edwards' original formulation because of the lack of explicit 4-dimensional symmetry in his transformations.

In a general inertial

frame with (w,x,y,z), we obtain precisely the same form as (C.43) in Appendix C for the decay rate:

r(l-»2 + 3 + ... + N) = » - „ ,c|(flSli)| J

w

where w=bt in a general inertial frame.

d3X2d3p2^d3XNd3pN|

(2jl/i)3

^ ^ ^

(2jt/i) 3

This definition of decay rate has the

dimension of inverse time because of the presence of the constant c in (17.71). The decay l i f e t i m e T

is given by x=l/r(l-*2+3+...+N).

To illustrate the

calculation of a decay rate, let us consider a simple example, i.e., the muon decay

Extended Relativity

261

M'~(Pi)~*e~(P2) + v^(p3) + v e (p4) with the usual V-A coupling.

Following the

steps in Appendix C, the muon lifetime x can be calculated and the result is 6

i = r ( l -» 2 + 3 + . . . + N) x

J-JiLEl«Lei 1 -p 2 - P3 -p 4 )2:|M sc | 2 . P01 J P02 P03 P04

^

(17.72)

'

Everything to the right of 1/poi in (17.72) is invariant

under the

extended transformation so that the "decay lifetime" x is indeed dilated by the usual y factor:

x oc ^ p 2 + m 2

=

p Q i a Y_

(17.73)

One should keep in mind, however, that it is really the decay length xc which is measured in the laboratory.

Such a "decay time" is directly related to the

lightime, rather than Reichenbach's time, as one can see from (17.71).

262

A Broader View of Relativity

References 1.

C. N. Yang, Phys. Today, pp. 42-49, (June, 1980).

2.

C. N. Yang, Science 127, 565 (1958).

3.

Leonardo Hsu, Jong-Ping Hsu and Dominik A. Schneble, Nuovo Cimento B,

4.

W. F. Edwards, Am. J. Phys. 3 1 , 482-489 (1963); H. Reichenbach,

111, 1299 (1996). Philosophy

of Space

and

Time

(Dover, New York, 1958).

The

Edwards'

transformations of space and time (17.8) have been rederived and discussed by several authors. See, for example, J. A. Winnie, Phil. Sci. 37, 81-99 and 223-238 (1970); Yuan-zhong Zhang, Special Relativity Foundations

and Its

Experimental

(World Scientific, Singapore, 1997) pp.75-101, and Gen. Rel.

Grav. 27,475 (1995). For a discussion of the one-way speed of light and its experimental tests, see, for example, A. A. Tyapkin, Lett. Nuovo Cimento 7, 760 (1973); B. Townsend, Am. J. Phys. 5 1 , 1092 (1983); R. Weingard, Am. J. Phys. 5 3 , 492 (1985); and Yuan-zhong Zhang, Special Relativity Experimental 5.

Foundations

and Its

(World Scientific, Singapore. New Jersey, 1997).

For a detailed discussion of common time and its implications

in

4-dimensional symmetry framework, see J. P. Hsu, Nuovo Cimento B74, 67 (1983); 88B,140(1985); 89B, 30 (1985); Phys. Lett. A97, 137 (1983); J. P. Hsu and C. Whan, Phys. Rev. A38, 2248 (1988), Appendix. 6.

See, for example, J. D. Bjorken and S. D. Drell, Relativistic

Quantum

Mechanics (McGraw-Hill, New York, 1964) pp. 261-268 and pp. 285-286; J. J. Sakurai, Advanced

Quantum

Mechanics

(Addison-Wesley, Reading, MA,

1967) pp.171-172 and pp. 181-188.

Extended Relativity

263

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c.

The Role of the Principle of Relativity in the Physics of Accelerated Frames

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18. The Principle of Limiting Lorentz and Poincare" Invariance Every generalization is a hypothesis.... It is clear that any fact can be generalized in an infinite number of ways, and it is a question of choice.

The choice can only be guided by considerations of

simplicity. H. Poincare, Science and 18a.

Hypothesis

An answer to the young Einstein's question and its

implications

Ever since the precise formalisms for mechanics and electrodynamics i n inertial frames were discovered and understood, physicists have attempted to discover the exact laws of physics in non-inertial frames. "Is it conceivable

that the principle

of relativity

also holds for

systems

which are accelerated relative to each other?" This was the question the young Einstein asked in his 1907 paper, two years after he created special relativity for inertial frames. 1 He said that this question must occur to everyone who has followed the applications of the relativity principle.

He considered two reference frames F^ and F2. The frame F^ is

accelerated in the +x direction with a constant acceleration g, and the frame F2 is at rest in a homogeneous gravitational field which gives all objects an acceleration of -g in the x-direction.

Einstein

relativity

to the case of the "uniformly

accelerated

complete

physical

corresponding

equivalence

constant

of a homogenenous

acceleration

of a reference

extended

the principle

of

frame"

by assuming

the

gravitational

field and the

frame.1

However, such an idea of extension does not seem to enable physicists to derive an exact generalization of the Lorentz transformation for frames with uniform accelerations or rotations.

One might think that perhaps this is

partially due to the fact that the essential 4-dimensional symmetry of the

267

Lorentz transformation was not understood at that time. But even today, although special relativity can be used to analyze situations in which particles are accelerated, the exact operational meanings of constant linear accelerations and uniform rotations of non-inertial frames are still not completely clear.2 There is a difficulty in the conventional notion of constant acceleration. A constant acceleration g implies a velocity vj = gti as measured in an inertial frame, in which the time ti is not limited in any way. Therefore, this velocity vj will eventually exceed the speed of light after a sufficiently long time ti, however small g may be. This violates special relativity and experimental results of particle kinematics. Therefore, the exact operational meaning of "constant linear acceleration" (CLA) cannot be satisfactorily based on the conventional concept of acceleration dvi/dti = constant in an inertial frame. Nevertheless, by considering small gt^ Einstein in 1907 was able to obtain two results related to time in accelerated frames:

(A)

T(x)-T(0)fl + ^ J .

(B)

The Maxwell equations in a uniformly accelerated frame ^ have the same form as in an inertial frame, but with the velocity of light c replaced by

c (l + £f).

(18.1)

(18.2)

Based on the equivalence principle, result (A) is interpreted to be equivalent to the metric tensor component g0o~(l+2gx/c2) for the spacetime associated with an accelerated frame, and is consistent with the gravitational red shift. On the other hand, result (B) was not completely satisfactory. The 4-dimensional spacetime symmetry of relativity was introduced by Poincare and Minkowski,3 and is one of the most thoroughly tested symmetry principles of the 20th century. It is the mathematical manifestation of the first postulate of special relativity, i.e., the symmetry or invariance of physical laws. The Lorentz and Poincare invariance (or the 4-dimensional symmetry of the

268

Principle of Relativity in the Physics of Accelerated Frames

Lorentz and Poincare groups) is fundamental and is extremely powerful helping us to understand physics.

in

4

Although there is a definite non-equivalence between an inertial and a non-inertial frame, we shall demonstrate in this chapter that the question raised by Einstein in 1907, namely whether "the principle of relativity also holds for systems which are accelerated relative to each other" can be answered based on the following two reasons: (A) The properties

and characteristics

of any accelerated frame

F(w,x,y,z)

must smoothly reduce to those of an inertial frame Fi(wj,Xj,yj,Zj) in the limit of zero (B) The principle

acceleration.

of relativity

under a coordinate

states that the laws of physics are transformation

Logically, this is equivalent

between inertial

to the statement

physics in inertial frames must display the symmetry

of the Lorentz and the Poincare

invariant

frames.

that the laws of four-dimensional groups.

Thus, it is natural and worthwhile to investigate spacetime transformations for accelerated frames on the basis of the principle of "limiting Lorentz and Poincare invariance" or "limiting 4-dimensional symmetry". states

that

4-dimensional acceleration.

the laws of physics symmetry

in non-inertial

frames

This

must

principle

display

the

of the Lorentz and Poincare groups in the limit of zero

As we shall

later

demonstrate in

section

19d, the

only

relativistically consistent definition for "constant-linear-acceleration (CLA)" is one in which a particle undergoing such an acceleration experiences a constant change of its relativistic momentum per unit time. Equivalently, if a particle moves with a constant linear acceleration, then the change per unit length of its energy is constant as measured by observers in any inertial frame.

Thus,

accelerations of reference frames can be investigated on the basis of a purely kinematical approach, independent of the gravitational field.

Physical results

can be obtained without using gravity as a crutch, in contrast to Moller's approach to deriving accelerated transformations of spacetime based on general relativity. 5 The "4-dimensional" spacetime (w,x,y,z) of n o n - i n e r t i a l frames is, of course, no longer the same as the relative spacetime of inertial frames in taiji

The Principle of Limiting Lorentz and Poincare Invariance

269

relativity. This is obvious because there is no relativity or reciprocity between an inertial frame and a non-inertial frame. In other words, such a spacetime (w,x,y,z) cannot have the 4-dimensional symmetry of the Lorentz and the Poincare groups. It is a more general 4-dimensional spacetime which includes the 4-dimensional spacetime of inertial frames as a special case when the acceleration approaches zero. To avoid confusion, we will call the more general spacetime "taiji spacetime." The evolution variable w of non-inertial frames will remain the "taiji time." Thus, taiji spacetime contains both non-relative spacetime and the relative spacetime of taiji relativity. As usual, we use non-relativistic to describe properties or quantities in Newtonian physics; but we shall use non-relative to denote properties or quantities associated with non-inertial frames. We will use the subscript I to denote quantities that are measured by observers in an inertial frame. Finally, taiji spacetime for non-inertial frames is not a general covariant theory (such as general relativity) because only a particular type of coordinates can be used in an accelerated transformation, just as only Cartesian coordinates can be used in the Lorentz transformation. Based on this limiting 4-dimensional symmetry principle, the answer to Einstein's question is affirmative in the sense that we can obtain a "minimal generalization" of Lorentz transformations for non-inertial frames with a constant-linear-acceleration (CLA). "Minimal generalization" means that the resultant equations involve a minimal departure from the simple case with zero acceleration. (See (18.8) below.) The 4-dimensional symmetry of the Lorentz and the Poincare groups is the essence of the Lorentz and Poincare invariance or the Poincare-Einstein principle of relativity. It must be stressed that "limiting Lorentz and Poincare invariance" is simply Lorentz and Poincare invariance applied to non-inertial frames in the limit of zero acceleration. We note that it is only the first postulate of relativity theory that is applied to non-inertial frames. The reason is because the speed of light (measured in any units) cannot be a universal constant in non-inertial frames. It is interesting and gratifying that the principle of limiting Lorentz and Poincare invariance has further physical implications. As we shall see later, it can also lead to transformations of spacetime for frames with general-linearaccelerations and rotations, and reveal which constants are fundamental in both inertial and non-inertial frames.

270

Principle of Relativity in the Physics of Accelerated Frames

18b. Generalizing Lorentz transformations from inertial frames to accelerated frames Since 1909, accelerated motion and transformations for accelerated motion have been discussed by many. 5 - 8 The results not been satisfactory because they lacked the limiting Lorentz and Poincare invariance. In section 18d and chapter 19 below, we show that the idea of limiting Lorentz and Poincare invariance leads to a coordinate transformation between an inertial frame and a frame with a constant acceleration with two parameters, the constant acceleration and the initial velocity, that reduces to the coordinate transformations of taiji or special relativity theory in the limit of zero acceleration. Let us first briefly consider the history of attempts to generalize the Lorentz transformations to accelerated frames. Soon after the introduction of special relativity, there were a number of attempts to define a transformation between the evolution variable in an inertial frame and the evolution variable in a frame with a constant linear acceleration. In 1907, through an ingenious trick of clock synchronization with the help of three reference frames, Einstein was able to obtain an important result (18.1) for clocks at different positions in a CLA frame F with small accelerations in the x direction:1

tx-tojl + ^ l

F=F(ct,x,y,z).

(18.3)

However, this approximate result is inadequate to reveal the relationship between time in an inertial frame and that in a CLA frame. In 1909, Born solved the relativistic equation of motion, dpj/dt] = F, using a constant force F=mg in an inertial frame. Using the relativistic momentum for Pi, he obtained6

(

2) Xi+ —

2

-N -

( 2\

cz .8,

The Principle of Limiting Lorentz and Poincare Invariance

271

which is a hyperbola in the Xjtj plane. The above results still do not lead to an exact coordinate transformation between an inertial frame and a CLA frame. More ambitious attempts to develop full

coordinate

transformations

between inertial and non-inertia frames came later. In 1943, Moller 5 obtained a spacetime transformation between an inertial frame Fj(ctj,Xi,yj,Zi) and a CLA frame F(ct,x,y,z) moving with a constant acceleration "a" in the Xj (or x) direction:

• u at

cti

sinn —,

X[ ••

c

, at cosh c

based on Einstein's vacuum equation R(iv=0

an

&

(18.5)

, yi a

d a postulated time-independent

metric tensor of the form, g| XV =(goo(x),-1,-1,-1). However, his transformation involved only one parameter, i.e., the constant acceleration, and hence it is of limited applicability because it does not reduce to the Lorentz transformation with a non-zero constant velocity in the limit of zero acceleration. In 1972, Wu and Lee 7 employed a kinematical approach to derive a uniformly

accelerated transformation

in the x-direction

by

assuming

a

time-independent metric tensor g^v(x) = ( g 0 0 ( x ) , - l , - l , - l ) and a local Lorentz contraction of length. Their results turned out identical to those of Moller. Later, Hsu and Kleff10 used Wu-Lee's kinematical approach to obtain a generalized transformation of time from an inertial frame to a CLA frame F(ct,x,y,z) with an arbitrary initial velocity v 0 and a constant acceleration "a" i n the x direction. We define w=ct in the CLA frame to facilitate comparison with Moller's and Wu-Lee's work. This may be called the generalized Moller-Wu-Lee (MWL) transformation.

One has the following transformation of the evolution

variable between an inertial frame Fj and a CLA frame F,

s i n h p a l + tanh- 1 ?,, ] - & £ - ,

cti

aYo

xi =

272

2 ^ c o s h p ^ + tanh_1p0|- —, a

Yo

yi=y, zI=z

Principle of Relativity in the Physics of Accelerated Frames

(18.6)

B = - = t a n n i c + tanh" 1 6 0 c { c

Yo

T^Hpi"

(18.7)

One can verify that eq. (18.6) reduces to (18.5) in the limit of zero initial velocity, B0 -> 0. Moreover, (18.6) reduces to the usual Lorentz transformation, ctj = Y0[ct+R0x], in the limit of zero acceleration, a-^0. The generalized MWL transformations (18.6) will be derived and discussed in the next chapter. However, based on the limiting 4-dimensional symmetry (see chapter 19), one can obtain yet a n o t h e r

transformation

Fi(wi,xi,yi,zi) and a CLA frame F(w,x,y,z):

between

an

inertial

frame

11

W!=YB «oYo

«oYo

(18.8) x,=Y

yi = y. «oYo;

Z] = z ;

«oYo

where one may write, if one wishes, wj = ctj, and

Yo : 1-P

•Jl^

Z

We call this the Wu transformations for CLA frames and it will also be discussed in the next chapter.

Note t h a t B in (18.8) has the usual form of a linear

function of the evolution parameter w, B = aow+B0> in contrast to the more complicated function of 6 = v/c = tanh(gt/c + tanh-ip,,) in (18.5) and (18.6). I n this sense, (18.8) is formally a minimal generalization of the

classical

transformations for accelerated frames.

The Principle of Limiting Lorentz and Poincare Invariance

273

18c.

Physical time and 'spacetime clocks' in linearly

accelerated

frames The limiting 4-dimensional symmetry naturally dictates the smooth connection between the physical time of inertial frames and that of accelerated frames in the limit of zero acceleration.

Since lifetime dilations of particles

with constant velocities can be described by the physical time in inertial frames, the corresponding "physical time" in accelerated frames must be able to describe, say, the accelerated "lifetime dilation" or "decay-length dilation" of an accelerated unstable particle in flight.

In this sense, the "time" or "evolution

variable" in both inertial and non-inertial frames should have equal

physical

significance. Thus, it is important first to define properly what the "physical time" is in accelerated frames and how one can realize it operationally through the setup of a clock system.

This is a very difficult problem in a general n o n - i n e r t i a l

frame, especially in rotating frames, because the speed of light is no longer constant so that the usual synchronization procedure based on light signals cannot be applied. Since there is no constant speed of light c in an accelerated frame F from the operational viewpoint, one can only denote the 4-coordinate of an event by (w,x,y,z) in general, according to the principle of limiting symmetry.

4-dimensional

Again, w has the dimension of length and plays the role of the

evolution variable. We avoid introducing the constant c = 299792458 cm/s into the formalism of physics in non-inertial frames. or logically wrong unnecessary.

to do so, defining

Although it is not physically

w=ct in

n o n - i n e r t i a l frames

is

9

As discussed in chapters 2 (sec. 2c.) and 7 (sec. 7d.), the spacetime of an inertial frame can be pictured as a rigid grid of meter sticks and identical clocks that have been synchronized assuming the universality of the speed of light. For simplicity, let us use the term "spacetime

clock" to denote a clock (on the

grid) shows both the time wj and the position n of the clock, i.e., (wi,xj,yi,zi). One can always use the spacetime clock an event to record the spacetime coordinates of that event. Since the spacetime transformation between an inertial and an accelerated frame is non-linear, one can no longer picture the spacetime coordinates of a CLA frame as a uniform grid with spacetime clocks. To be more specific, let us consider the inverse of the Wu transformation (18.8):

274

Principle of Relativity in the Physics of Accelerated Frames

w=

Wi+p0/a0y0_Po. , COXJ+1/YO

wj.ctj,

a0

(18.9) 2

w

X = V(xI+l/Yoa0) -( I+Po/aoYo)

2

7.

y = Yl >

z = Zj.

«oYo

With this transformation in hand, the spacetime clocks in the CLA frame can be synchronized without any further information. This is possible because any "clock" (in the sense of a device which shows the time and position) can be adjusted to run at an arbitrary rate and show an arbitrary time and position. For example, our F "clocks" could be some kind of computerized devices which have the capability to measure their position rj in the Fi frame, to obtain wi = cti from the nearest Fi clock, and then to compute and display w and r=(x,y,z) using (18.9) [with known velocity p0 and acceleration aQ] on some readout. As a result, if the clocks in the CLA frame are uniformly spaced on a regular grid, the differences in their time readings will not be uniform. However, this grid of computerized spacetime clocks will automatically become the more familiar Einstein clocks of inertial frames, provided that w=ct and w'sct', when the acceleration a 0 in (18.9) approaches zero. This indicates that the coordinates (w,x,y,z) in (18.8) for a CLA frame play the same role and have a similar physical meaning to the spacetime coordinates in the transformations for inertial frames. In the following discussions, we shall use (w,x,y,z)

And

(wi,xi,yi,zi)

(18.10)

as coordinates of non-inertial and inertial frames respectively. For simplicity, w and wi are called the "time." 18d. Meller's gravitational transformations

approach to

accelerated

It appears that physical phenomena in a constant-linear-acceleration (CLA) frame have not been thoroughly investigated. Although there have been lifetime dilation experiments for particles in uniform circular motion, these

The Principle of Limiting Lorentz and Poincare lnvariance

275

results cannot be applied to the case of CLA motion in which a particle's speed changes.

Also, we still do not have a satisfactory transformation between an

inertial frame and a CLA frame which can be smoothly and naturally connected to the Lorentz transformation when the accelerated frame becomes an inertial frame.

Moller used Einstein's vacuum equations Rik = 0 and postulated a

time-independent goo and g n = g22 = g33 = - 1 , ds 2 = goo(x)dw2 - dr 2 , to obtain gOo(x)= (1+gx/c 2 ) 2 and a transformation between an accelerated frame and an inertial frame. 6

But his accelerated transformation

cannot be

smoothly

connected to the Lorentz transformation in the limit of zero acceleration. In this section, Einstein's equation R^ = 0 is used as a guiding principle to obtain a transformation between an inertial frame and a CLA frame. Einstein's

covariant

equation

holds

for

any

Although

coordinates, the

Lorentz

transformation prefers the Cartesian coordinates. Thus, the natural assumption of smooth connection between a CLA frame and an inertial frame dictates that the CLA transformation be expressed in quasi-Cartesian coordinates w h i c h become Cartesian in the limit of zero acceleration.

The reason for using R^ = 0

is suggested by a heuristic view that the 'inertial force' of accelerated frames and the 'gravitational force' may be considered as having been "unified" by Einstein's equation. conditions:"

Nevertheless, these two forces satisfy different

"boundary

Namely, in contrast to the gravitational force, the inertial force

does not vanish at spatial infinity, and the transformations for accelerated frames should reduce to the Lorentz transformation when inertial

forces

vanish. Let us first consider an inertial frame Fi(wi,xi,yi,zi), and a CLA frame F(w,x,y,z) moving with a constant acceleration along the x-axis. Based on the preceding discussions, it is natural to assume that ds 2 takes the form ds 2 = c2dt,2 - dx 2 - dy 2 - dz,2 = g0o(x, w)dw 2 + g n (x)dx 2 + g 22 (x)dy 2 + g 33 (x)dz 2 ,

(18.11)

where g 00 is assumed, in general, to be a function of w and x. The time tj is shown by the Einstein clocks in the inertial frame Fj. Although Rik= 0 holds for arbitrary coordinates, we postulate the metric (18.11) so that ds 2 and t h e resultant transformations are consistent with both Einstein's vacuum equation

276

Principle of Relativity in the Physics of Accelerated Frames

Rik=0 and the new "boundary condition" of limiting four-dimensional symmetry when the acceleration vanishes. Since the CLA frame F moves along the x-axis, we look for axially symmetric solutions with g22 - g33 = -Y 2 (x)

(18.12)

and where all metric tensors components are functions of x, except that goo may be a function of x and w. These physical properties of goo(XiW) are crucial to the new CLA transformation; and will be determined later. Now we can calculate Christoffel symbols G'jk = gim(3kgmj + djgmk - dmgjk) and the Ricci tensor Rjk = S m ^ k + dkC^im +GnikGmnm- G^n^km. where 3k = a/axk, k=0,l,2,3. The equations Rn = 0,i=0,l,2, lead to

a | W - a x W | ^ - - 2 ^ =0, X

Y j

a x =—,

(18.13)

dx

a|w+2a|Y_dsx(23xY+^w| = 0 W

afo Y

Y

3

X

x Ya x X

YX

Y

W

, ( a x Y f , 3 X Y^W _ Q Y YW

(18 15)

respectively, w h e r e Y is given b y (18.12) a n d W 2 = g 0 0 (x, w),

X2 = - g u ( x ) ;

W = W!(x)W 2 (w).

(18.16)

Here, R33 = 0 gives t h e s a m e e q u a t i o n as (18.15) a n d o t h e r c o m p o n e n t s of Rik

vanish identically. If d2xY = 0, equations (18.13)-(18.15) lead to an exact solution WL -

, f3 , V f i x + f0

X=f 2 Vfix + f0,

Y = f1x + f0>

The Principle of Limiting Lorentz and Poincare

(18.17)

Invariance

277

where fi, f2 and f3 are constants.

The quantity W2(w) in (18.16) is arbitrary

because it cannot be determined by Einstein's equation. Physically, one expects that the metric tensor g22 should satisfy -g22 = Y2 = 1 rather than Y 2 = (fix + fo) 2 , since there is no motion along the y-axis.

Furthermore, the accelerated

transformation based on this solution cannot be smoothly connected to the Lorentz transformation (i.e., it does not satisfy the "limiting four-dimensional symmetry" or integrability conditions in (18.23) below.) Therefore, the solution (18.17) with f!* 0 is not physically meaningful.

The case f1= 0 in (18.17) is

trivial and uninteresting because it is equivalent to that of zero acceleration. Let us concentrate on the nontrivial case dxY = 0 and axW = 0. We have the solution Y=l,

(18.18)

which satisfies the boundary conditions g22(0) =g33(0) = - 1 at the origin.

From

(18.13) and (18.18), we deduce a general relation between Wi(x) and X(x):

dW (x)

i =fX(x), dx

(18.19)

where f is a constant of integration. The function Wi(x) can be determined if X(x) is given or postulated. However, the time-dependent part of goo, i.e., W2(w), still cannot be determined by Einstein's equation, just as in the previous case because axY = 0. Thus we have seen that Einstein's covariant equation by itself does not lead to a specific form for X(x), Wj(x) and W2(w).

One must have

specific functional forms for X(x) and W in order to have finite transformations between a linearly accelerated frame and an inertial frame.

Moller postulated

W=Wi(x) and X(x)=l in (18.16) (i.e., time-independent goo and g n =g22 =g33 =-1) to obtain goo(x) = (1+gx/c 2 ) 2 and the transformation between an accelerated frame and an inertial frame given in (18.5).6

278

Principle of Relativity in the Physics of Accelerated Frames

18e. Accelerated transformations with the Lorentz and Poincar£ invariance

limiting

Let us now consider the implications of the limiting Lorentz and Poincare invariance for the transformation equations between a CLA frame F and an inertial frame Fj. Suppose that a CLA frame F moves along parallel x and xi axes and that the origins of F and F[ coincide at the taiji-time w = wi = 0. We postulate that the invariant infinitesimal intervals for a CLA frame F and an inertial frame Fjare given by ds2 = W2(w, X)dw2 - dx2 - dy2 - dz2 = dw2 - dx,2 - dy,2 - dz,2 .

(18.20)

The principle of relativity dictates that W(w,x) must approach 1 in the limit of zero acceleration. This is the requirement of the limiting Lorentz and Poincare invariance (4-dimensional symmetry). The presence of the function W(w,x)*l implies the physical non-equivalence of F and Fi. The local relation between F(xM-) and Fj( x p may be written in the form, dw] = v(Wdw + pdx), (18.21) dxj = v(dx + fSWdw),

dyj = dy,

dzi = dz ,

where y and (5 are functions of the time w in general, and

Y--T=i— •

(18.22)

The local transformation (18.21) preserves the invariance of ds2 in (18.20). In order for F(w,x,y,z) to reduce smoothly to an inertial frame in the limit of zero acceleration, there must exist a global transformation related to (18.21). In other words, the limiting 4-dimensional symmetry also dictates that the two unknown function W(w,x) and p(w) must satisfy the following two integrability conditions for the differential relations in (18.21):

The Principle of Limiting Lorentz and Poincare Invariance

279

a(YW) _ a(vp)^ ax aw '

_3L=5(YPW)_ aw ax

and

( l g 2 3 )

By separation of variables, W(w,x)=WwWx, the two equations in (18.23) lead to the same relation dWx=J^dp_ dx Ww dw

1

where ki is a constant. Thus we have two solutions W x =k 1 x + k 2 ,

w

and

_d?/dw kid-p 2 )

(18.25)

(lg26)

The implication of the principle of limiting 4-dimensional symmetry is as follows: the solution of the differential equation (18.26) depends on the physical properties of either p or Ww. If the "velocity function" p is known, then Ww can be determined, and vice versa. The principle of limiting 4-dimensional symmetry by itself cannot uniquely determine the function W(w,x). In general, there are infinitely many solutions for (18.24). We observe that the present case resembles the situation in which gauge symmetry cannot determine uniquely the electromagnetic action, 12 and one must further postulate a minimal electromagnetic coupling. In the present case, we postulate a minimal generalization of the Lorentz transformation. The minimal generalization of the Lorentz transformation is to give up the untenable relation vi = gti + vi0 in an inertial frame Fi (as discussed in section 18a,) and to retain the velocity function p as the usual linear function of time w in the CLA frame, p = o0w + p 0 .

280

(18.27)

Principle of Relativity in the Physics of Accelerated Frames

It follows from (18.25) and (18.26) that we have a time-dependent metric tensor g00(w,x)=W2, where

W = WWWX = v2(Vo2 +«0x) = {y'°2 + " ° X ) , (1-P2)

(18-28)

and in which the constants k2 and ki in the product WWWX are determined by the limiting condition W(w,x) -»1 as a o -»0. However, there is another relatively simple generalization of the Lorentz transformation that can be obtained by postulating a time-independent metric tensor goo , i.e., g 0 0 =W x

or

Ww-1.

(18.29)

It then follows from (18.25) and (18.26) that, W x =k 1 x + k 2 ,

(18.30)

p = tanh(k!W + k 3 ),

(18.31)

with parameters ki, k2, and k3 determined as follows. The constant k 3 in (18.31) can be determined by the initial condition that p = p0 when time w = 0, i.e., k 3 = tanh _1 p 0 . Furthermore, since p is equal to p0 when the acceleration approaches zero, ki in (18.31) should be proportional to the constant acceleration, although ki may also depend on p0, as shown in eq. (18.35) below. Finally, since Wx must be 1 if the acceleration (and thus ki) vanishes, we see that k 2 =l. Relations (18.30) and (18.31) are now determined as follows: Wx = l + k!X,

and

p = tanh(kxw* +tanlr 1 p0),

(18.32)

where we have denoted the time as w* in order to distinguish it from w in (18.27). In this case, the simple relationship between the velocity and time variables in (18.27) has been replaced by a more complicated relationship.

The Principle of Limiting Lorentz and Poincare Invariance

281

Thus, we do not consider (18.32) to be a minimal generalization. we will also discuss the

solution (18.32) because the

Nevertheless,

assumption of a

time-independent metric tensor was made by Moller and also by Wu and Lee, 6 ' 7 . Equation (18.32) leads to a generalization of the transformation obtained by them to the case of non-zero initial velocity, although it is based on entirely different considerations. 1 0

282

Principle of Relativity in the Physics of Accelerated Frames

References 1.

A. Einstein, Jahrb. Rad. Elektr. 4, 411 (1907). See also A. Pais, Subtle is the

2.

A. P. French, Special

Lord..., (Oxford Univ. Press, 1982) pp. 179-183. Relativity

pp.l54.V. Fock, The Theory

(Norton & Company, New York, 1968)

of Space Time and Gravitation

(Pergamon,

London, 1958) pp. 369-370; T. Y. Wu, private correspondence, (1997). 3.

H. Poincare, Rend. Circ. Mat. Pal. 2 1 , 129 (1906); H. Minkowski, Phys. Zeitschr. 10, 104(1909).

4.

C. N. Yang, Physics Today, June 1980, pp. 42-49; P. A. M. Dirac, Sci. Am. 2 8 , 48 (1963).

5.

C. Moller, Danske Vid. Sel. Mat-Fys. 20, No. 19 (1943); See also The Theory of

6.

M. Born, Ann d. Phys. 30, 1 (1909); A. Sommerfeld, Ann. d. Phys. 33, 670

7.

Ta-You Wu and Y. C. Lee, Int'l. J. Theore. Phys. 5, 307-323 (1972); Ta-You

Relativity (Oxford Univ. Press, London, 1952), pp. 253-258. (1910). Wu, Theoretical

Physics, vol.4, Theory of Relativity

(Lian Jing Publishing

Co., Taipei, 1978) pp. 172-175, and references therein.

The authors also

made an exact calculation of the clock paradox problem, including the effects of linear accelerations and decelerations. 8.

Jong-Ping Hsu and Leonardo Hsu, Nuovo Cimento 112, 575 (1997) and Chin. J. Phys. 3 5 , 407 (1997).

This is called the Wu transformation to honor

Ta-You Wu's idea of a kinematical approach to finding

an accelerated

transformation in a spacetime with vanishing Riemann curvature tensor. 9.

J. P. Hsu and Leonardo Hsu, Phys. Lett. A 196, 1 (1994); Leonardo Hsu and Jong-Ping Hsu, Nuovo Cimento B, 111, 1283 (1996).

10.

J. P. Hsu and S. M. Kleff, Chin. J. Phys. 36, 768 (1998); Silvia Kleff and J. P. Hsu, JingShin Theoretical Physics Symposium

in Honor of Professor

Ta-You

Wu (Editors J. P. Hsu and L. Hsu, World Scientific Singapore, 1998), pp. 348-352. 11.

J o n g - P i n g Hsu and Leonardo Hsu, Nuovo Cimento B 1 1 2 , 575 (1997); Leonardo Hsu and Jong-Ping Hsu, Nuovo Cimento B 112, 1147 (1997). See section 19b for derivations and discussions of the Wu transformations.

12. J. J. Sakurai, Invariance

Principles

and Elementary

Particles,

Princeton

Univ. Press, 1964) p. v, pp. 3-5, and p.10.

The Principle of Limiting Lorentz and Poincare Invariance

283

19. Extended Lorentz Transformations for Frames with Constant-Linear-Accelerations 19a. Generalized

Moller-Wu-Lee

transformations

We are now in a position to generalize the Lorentz transformations to reference frames with constant linear accelerations. Substituting (18.32) in (18.21), one can carry out the integration and obtain the relations wj = x +k— s i n h ^ w * + tanh 1 p 0 ) + k 6 ,

J

X! =fx + — cosh(k1w* + tanh- 1 p 0 ) + k 7 ,

Yi=y.

(19.1)

zi=z-

Using the usual initial condition that xi = x = 0, when wj = w* = 0, the constants of integration kg and kj in (19.1) can be determined in terms of ki k

6

= ^ , ki

k7=^a.

(19.2)

kj.

In order to determine the role of a constant acceleration in (19.1) and to compare it with the transformation obtained by Nfoller and by Wu and Lee,1-2 we follow the approach of Wu and Lee and impose the boundary condition that the velocity p is related to the constant acceleration a* by p = a*wi+pc at xi = 0.2 From (18.32), (19.1) and (19.2), ki is found to be ki=Yo«*,

(19.3)

because

284

p_

sinhfow'+tanh-1?,) _ w ^ P o / k ! . cosh(k 1 w*+tanrr 1 p 0 ) xi+Yo/ki

^

^

for

X]

.0.

Thus, the generalized M0ller-Wu-Lee (MWL) transformation for the frames Fi and F with non-zero initial velocity p 0 is given by

Wj = x + sinh(y 0 a * w * + tanh V YoO *)

1

p0)- — , a *

xj = x +

1

p0)

cosh(Y0a * w * + tanh

,

(19.4)

Vo« yi = y.

zi=z-

In the limit of zero acceleration a*-»0, transformation (19.4) indeed reduces to a form with the four-dimensional symmetry of the Lorentz group w, = Yo(w* + p 0 x),

X! =Y0(x + p0w*),

yi = y,

z\ = z,

(19.5)

where sinh(tanh _ 1 p0) = Y0Po>

an

d

cosh(tanh

1

p 0 ) = y0-

Furthermore, when p 0 approaches zero the accelerated transformations (19.4) reduce to the Moller-Wu-Lee transformations (18.5) with a/c 2 = a*, w = w* and cti = Wi. The result (19.4) is the same as (18.6) with a/c 2 = a*, w = w* and ct^Wi. 3 The inverse transformations of (19.4) are found to be -tanh-1 Vo" I

• tanh" 1 p0 Xj + 1 / a*

X = T/(Xi + l / a * ) 2 - ( w I + p 0 / a * ) 2

*

(19.6)

Yo"

yi = y,

zi=z-

Extended Lorentz Transformations

285

Differentiation of the generalized MWL transformation (19.4) gives dwi = y(Wxdw* +pdx), dxt = Y(dx + pWxdw*), dv[ = dy, dzj = dz. Wx = 1 + Yoa*x>

(19.7)

P = tanh(y0a*w* + tanh" 1 p 0 ), y = - ? =

{l-V2

It follows from (19.7) that ds 2 = dw2 - dx2 - dy 2 - dz 2 = w|dw* 2 - dx2 - dy 2 - dz 2 = g*vdx»Mxv, (19.8) g V - (Wx2, - 1 , -1,-D,

Wx = 1+YO«*X,

where n, v = 0,1,2,3. As is the case in which Cartesian coordinates are the preferred coordinate system for the Lorentz transformations, the coordinates x^=(w*,x,y,z) with metric tensor gj^v given by (19.8) are the preferred coordinates for the generalized MWL transformation. The coordinates x^ cannot be arbitrary because the principle of limiting 4-dimensional symmetry dictates that the transformation (19.6) or (19.4) must reduce smoothly to the Lorentz transformation in the limit of zero acceleration. In a gedanken sense, both the position x and the time w can be operationalized using a grid of computers that determine their own position x and the time w= w* they should display using equation (19.6) and the values of xj and wj obtained from the nearest Fj clock. The propagation of a light signal in the CLA frame is described by ds=0, i.e.,

dr = W x = l + a*Y0x dw^

(19.9)

and is consistent with Einstein's approximate result (18.2) when one sets w*=ct, po=0 and a*=g/c2.

286

Principle of Relativity in the Physics of Accelerated Frames

For a particle at rest in F], we obtain the expression for its proper lifetime (or decay-length) w Ip =/dwi =/ds using (19.7) and (19.8) with dxi = 0 and yi = zj = 0, i.e., w* J W Ip = / d w * 1 | w

2 x

~ w* -(^)2=a*Yo/dw

1

J

cosh2(v0a* w*+tanh

• \ + X! |tanh(7 0 a * w * + tanh"1 p0) - p 0 ],

1

p0) (19.10)

where Xj is the fixed position of the particle. On the other hand, if a particle is at rest at x in F, its proper lifetime (or decay-length) is w* p = fdw* = (1/WX) f ds = (1/WX) fdw I [l-(dx I /dw 1 ) 2 ] 1 / 2 , where x is fixed, so that we have w

1

w*

p ~ (l + a * x ) Yo

I 7 /dw^l-P2 0

= —!— ftanh-1Z(x,wI)-tanh-1Z(x,0)l,

(19.11)

Vo1

where (w 1+ p 0 /a*)

Z(x,w,) = 1 /(x +

dX; dwj

l/Voa*)2

+

(w I + p o / a * ) 2

= p = tanh(Yoa * w * + tanh" 1 p0) = w '

+ P

°/a*.

X] +1 / a i

dx=0

The usual acceleration, d2xi/dwi2 = dp/dwj of a fixed point r in F, as measured in Fj, is d 2 xi_ (x + l/ Yo a*) 2 2 dw, [(x + l/ V o a*) 2 + (w I + P 0 /a*) 2 ] 3 / 2 '

(19.12)

which is clearly not a constant. However,

Extended Lorentz Transformations

287

* P dw I ^1 _ p2

Y

3^L_^lla_, dW! l + X Yo a*'

is a constant because x is fixed. where dpj/dtj is a constant. 19b.

Minimal The

This is consistent with Born's result in (18.4)

generalization

Wu

(19.13)

of

Lorentz

transformations:

transformations

For the minimal generalization of the Lorentz transformations (described in section 18e), we retain the usual relation p=aoW+p0. The postulate of limiting 4-dimensional symmetry and the minimal generalization lead to the interval (18.20) with W given by (18.28).

Substituting (18.28) into (18.21) and

integrating, one obtains the CLA transformation:

wI=Yp

1 x+-

"oYo

1

*I=Y x +

a

oYo

\ v

"oYo;

i=y>

zi=z;

wi=ct],

(19.14)

<*oYo

where the constants of integration, Pc/a0Yo and l/a0Yo> can be determined by the

initial

conditions

transformation.

4

x;=x=0 when wi=w=0.

We call (19.14)

the

Wu

One can verify that (19.14) indeed reduces to the Lorentz

transformation with w*-»w in the limit of zero acceleration. The inverse Wu transformation derived from (19.14) is (wi+p 0 /a 0 y 0 ) (a 0 X!+l/Y o )

pQ a0 (19.15)

2

x = -J(x I +l/Y o a o ) -(w I +p o /a 0 Y 0 )

2

-, a0Yo

y = yi,

z = zr.

Differentiation of (19.14) gives dw] = Y(Wdw + pdx), dx! = Y(dx + pWdw), dyj = dy, dz! = dz;

288

Principle of Relativity in the Physics of Accelerated Frames

(19.16)

(5 = a 0 w + p 0 ,

Y—i-i—,

W = 7 2 (Yo 2 +« 0 x).

(19.17)

The invariant infinitesimal interval is thus ds 2 = dw,2 -dx, 2 - d y 2 - d z 2 = g (Jv dx |i dx v ,

(19.18)

g, v = (W2, -1,-1,-1) = (Y4[VQ2 + ax] 2 ,-1,-1,-1).

The factor W in the metric tensor (19.18) and the differential form of the Wu transformation (19.16) may be called the Wu factor.

As with the generalized

MWL transformation, the coordinates (w,x,y,z) with the metric tensor g^ v given by (19.18) are the preferred coordinates for the accelerated Wu transformation. The time w in (19.14) and (19.15) differs from that in (19.4) and (19.6), of course. We shall discuss physical implications of this important difference in section 20e. The propagation of light is described by setting ds=0 in (19.18), so that t h e speed of light measured in terms of w in a CLA frame is now given by the Wu factor W: dr_ = W = Y 2 (YO 2 + « 0 X ) , dw

(19.19)

which is consistent with the result (19.9) and Einstein's result (18.2) for small velocities, p o ~0 and l p 2 l « l . At large velocities however, they have significant differences.

Such differences are related to the differences between the two

functions p(w*) in (19.7) and p(w) in (19.17). If a particle is at rest in Fi at the location (xi,0,0), (19.18) implies that its proper lifetime wi p is w I p =J dwj = / ds, i.e., wIp = / d w V w 2 - ( d x / d w ) 2 J i + ^ J w , o Uo )

Extended Lorentz Transformations

dn=0.

(19.20)

289

On the other hand, if a particle is at rest in F (r=constant), (19.18) gives the proper lifetime wp as 3/2 =

;dw = / — = o W Yo

il

1 +a

W]

dX; dwj

-/dw] ox o

Wi+P0/a0Y0 y(x + l / Y o a 0 ) 2 +(wi + P 0 / a 0 Y o ) 2

Po/aoVo •J(x + l / V o a o ) 2 + ( P o / a o Y 0 ) "

1

(19.21)

This can also be written as

W] •

x+[YP-YOPO]Yo« 0 /

P = a 0 w + p0.

(19.22)

These relations for the dilation of the lifetimes of unstable particles could be tested experimentally by measurements of the decay-lengths (or "lifetimes") of a particles undergoing a constant linear acceleration. 19c.

A comparison of the generalized MWL and Wu transformations

In the preceding discussion, we obtained two simple transformations between an inertial frame and a CLA frame on the basis of the principle of limiting 4-dimensional symmetry. One was the Wu transformation (19.14) which was obtained by assuming a minimal generalization of the Lorentz transformations preserving the usual linear relationship between acceleration and velocity, p = aQw + p0, in accelerated frames. The other was the generalized Moller-Wu-Lee (MWL) transformations (19.4), which were obtained by assuming the metric tensor ( g 0 0 , - l , - l , - i ) to be time-independent. This transformation has a more complicated and unusual relationship between the acceleration and velocity, p =tanh(-y0a*w* + tanh _1 p 0 ). Both of the evolution variables w and w* in these two transformations can be considered as a

290

Principle of Relativity in the Physics of Accelerated Frames

generalization of the 'time' in inertial frames. We have seen that the physical 'time' in CLA frames cannot be uniquely determined by the limiting 4-dimensional symmetry principle. Let us summarize the properties of the generalized MWL transformations and the Wu transformations, both of which are based on the limiting 4-dimensional symmetry principle and an additional assumption: (A) Generalized MWL transformations: Additional assumption: goo(x|i) =goo(x) g 0 o=[l + a*Yox]2,

f^-j

=»

=tanh(a*Vow%tanh-1p0).

(19.23)

(B) Wu transformations: Additional assumption:

p = a0w + p0 =

dx] dwj

goo(w,x) = [ v 2 ( Y - 2 + a 0 x)] 2 .

(19.24)

It is interesting to observe that time w* in the generalized MWL transformation (19.4) and time w in the Wu transformation (19.14) are related by the following relation:^ w = - ftanh(a*YoW*+tanh-1p0)-p0], a0l. J

a* = a 0 y 0 .

(19.25)

This relation between w and w* can be seen by comparing (dxi/dwi)x in (19.23) and (19.24). It follows that

Wxdw* = (l + a*v 0 x)M— d[tanh-1(a0w + p0)l = W(w,x)dw,

(19.26)

and

Extended Lorentz Transformations

291

ds2 = dw,2 -dr,2 = W2dw*2 -dr 2 = W(w,x)2dw2 -dr 2 .

(19.27)

Although both times w* in (19.4) and w in (19.14) reduce to the same "taiji time" (measured in the unit meter rather than second) for inertial frames in the limit of zero acceleration, a relevant question is: "Which physical time in constant-linear-acceleration frames is correct?" Since the physical time in inertial frames is directly related to the dilation of the "lifetime" or to the decay-length of an unstable particle decaying in flight, the same must hold also in CLA frames because of the limiting 4-dimensional symmetry principle. Therefore, the question of which physical time is correct for accelerated frames can only be settled by experiments. One way to answer this question is to measure the decay-length dilation to test the predictions of the Wu transformations and the generalized MWL transformations. We will discuss such a test in more detail in chapter 20. 19d. Four-momentum and constant-linear-acceleration of an accelerated particle Now let us demonstrate that, in the framework limiting 4-dimensional symmetry, a "constant linear acceleration" means a constant change in the kinetic energy pio of an object per unit length, as measured in an inertial frame. Consider the invariant action Sf for a "free particle" in a CLA frame (which may be associated either with the generalized MWL transformations with time w* or the Wu transformations with time w): b

wb

Sf = -Jmds = jL w dw, a

(19.28)

wa

ds 2 = W 2 d w 2 - d x 2 - d y 2 - d z 2 ,

Lw = -mJg^u»uv "

= -m^W2-^)2,

g^v = (W2,-1,-1,-1),

u"

= - ^ = (1, p1). dw

The function W in (19.28) may be either Wx = (l+a*vox) with w replaced by w* or W(w,x) = Y2(Y32+CX0X).

292

Principle of Relativity in the Physics of Accelerated Frames

As usual, the covariant momentum p; and the corresponding "energy" po (or the Hamiltonian H) with the dimension of mass are given by

-mrpx -mrpy -mrpz

Po

1

ap

w 'w 'w

UP1

= mrW = g 0 0 P u ,

^i.r

Pi = gikP = - P ' .

0

0

dW

p u = m —— = m — - ; ds ds

/ir»in\

(19.29)

dx1 dw

P2 = Px + Py + Pz

-\/l-p 2 /W 2

dX

i = l,2,3;

Thus, the covariant momentum 4-vector p ^ = (po,Pi,P2,P3) = (Po,-P) transforms like the covariant coordinate dx^=g^ v dx v . We have

Pro = Y ( ^ - - PPi),

Pn = Y(PI - ^2-),

P12 = P2>

Pi3 = P3; (19.30)

P f o - P f - f e o / W2^ - -p2^ m Now suppose a particle is at rest in the CLA frame (p = 0). From (19.30), p 0 = mW and p I0 = my. Thus, when r is fixed, we obtain

fdPio)

[ dxi Jx

3 = m y

^

(19.31)

dxi'

ma/(Yo +a0x);

mY0a*/(l+a*Y0x);

p = a0W + p0

p = tanh[a*Y0w* + tanh 1 p o ] .

The last relation involving a* is consistent with the constant acceleration given in (19.13). The result (19.31) is intimately related to the "limiting four-dimensional symmetry." The concept of linear and uniform acceleration of a particle can be

Extended Lorentz

Transformations

293

defined in the sense of (19.31), i.e., a constant change of a particle's "energy" pio per unit length, as measured in an inertial frame Fi. It is gratifying to see that this is precisely the type of acceleration that is realized in high laboratories.

Therefore, the theories with limiting 4-dimensional

energy

symmetry

already have partial experimental support. 19e.

Experiments on Wu-Doppler effects of waves emitted from accelerated

atoms

The new accelerated transformations

(19.4) and (19.14) can also be

experimentally tested by measuring a Doppler-type shift of wavelength of light emitted from an accelerated source.6

From eq. (19.30) one obtains the Wu

transformation of the covariant wave 4-vector k^ = p^/J between an inertial frame Fj and a CLA frame F: kI0=Y(k0/W-pk1), where

kI1=Y(k1-pk0/W),

k2=k3=0,

(19.32)

ki^= (kio,-ki ) and k^ = (ko,-k). As discussed earlier in section 10b, Jkio

and Jko are moving masses of the same photon measured by observers in Fi and F respectively. Suppose the radiation source is located at the origin of the F frame and is co-moving with that frame, r = 0.

Experimentally, it is difficult for

observers in the accelerated frame F to measure ko(rest) and ki(rest) for such a radiation source. Thus we express the shifts in terms of quantities measured by observers in the inertial frame (or laboratory) Fj. Using (10.14) as the approximate relation,, i.e. ko(rest) a kio(rest), and W(w,0) = Y2YO2, we obtain a Doppler-type shift of kio (related to the photon's moving mass or the difference between the atomic mass levels of the radiation source) and wavelength Xl

k I0 = k I0 (rest)

^ i ^

m2

(19.33) 1 _ 1 X: ^(rest)

(1 + p) YY-2

for waves emitted from a radiation source undergoing a CLA. These new results predicted by the Wu transformation are termed the Wu-Doppler shift. 5

294

Principle of Relativity in the Physics of Accelerated Frames

For an experimental test, it is more convenient to express 'velocities' in (19.33) in terms of distances. Suppose the radiation source (at r = 0) enters an accelerated potential at xt = 0 in Fj with an initial velocity p0 and emits radiation when it has been accelerated to the point x; = L with a velocity Pj. We have

^"te")

£i

= = 1

P V -vo 2 ( 1+La oYor 2

(19.34)

= Y0(l + a 0 Lv 0 ).

(19.35)

It follows from (19.33)-(19.35) that the wavelength shift is given by Ski =ki- X](rest) - ^(rest) a 0 L - ^ + 2 a 0 L

- i2 p^ 02

(19.36)

Also, according to (19.33), we expect the shift of mass levels of an atom at rest in a CLA frame to be 5M! = JkI0 - JkI0(rest) - JkI0(rest) /p2+2a0L

+ipo-

a

oL

(19.37)

The predictions (19.36) and (19.37) can be measured in the laboratory frame Fj using a method similar to that in the Ives-Stilwell experiment.''

Extended Lorentz Transformations

295

References 1.

C. Moller, Danske Vid. Sel. Mat-Fys. 20, No. 19 (1943); See also The Theory of Relativity (Oxford Univ. Press, London, 1952), pp. 253-258.

2.

Ta-You Wu and Y. C. Lee, Int'l. J. Theore. Phys. 5, 307-323 (1972); Ta-You Wu, Theoretical

Physics, vol. 4, Theory of Relativity

(Lian Jing Publishing

Co., Taipei, 1978) pp. 172-175, and references therein.

The authors also

made an exact calculation of the clock paradox problem, including the effects of linear accelerations and decelerations. 3.

J. P. Hsu and S. M. Kleff, Chin. J. Phys. 36, 768 (1998); Silvia Kleff and J. P. Hsu, JingShin Theoretical Physics Symposium

in Honor of Professor

Ta-You

Wu (Editors J. P. Hsu and L. Hsu, World Scientific Singapore, 1998), pp. 348-352. 4.

Jong-Ping Hsu and Leonardo Hsu, Nuovo Cimento 112, 575 (1997) and Chin. J. Phys. 35, 407 (1997).

5.

J o n g - P i n g Hsu and Leonardo Hsu, in Symposium

JingShin

Theoretical

Physics

in Honor of Professor Ta-You Wu (Ed. J, P, Hsu and L. Hsu, World

Scientific, Singapore, New Jersey, 1998) pp. 393-412. 6.

Jong-Ping Hsu and Leonardo Hsu, Nuovo Cimento B. 112, 1147 (1997); Chin. J. Phys. 35, 407 (1997).

7.

296

H. E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938).

Principle of Relativity in the Physics ofAccelerated Frames

20. Physical Properties of Spacetime in Accelerated Frames 20a.

A general transformation

for a CLA frame with an arbitrary

p(w)

Our discussions so far have been based on the principle of limiting 4dimensional symmetry which requires that all accelerated transformations reduce to the Lorentz and Poincare transformations in the limit of zero acceleration and hence, that physical laws in an accelerated frame F can be obtained by transforming the laws of physics from their commonly known forms in inertial frames to an accelerated frame F.l This suggests that the spacetime coordinates of accelerated frames are as meaningful as those of inertial frames. In this chapter, we illustrate these properties and the distortions in the spacetime of accelerated frames graphically, including singularities and horizons. Physical implications of various accelerated transformations for the lifetime and decay-length dilations of unstable particles undergoing accelerated motion are calculated in section 20e.2 A generalized Lorentz transformation must be consistent with the fact that any accelerated frame reduces to an inertial frame in the limit of zero acceleration. This is a natural, basic and necessary requirement from a physical viewpoint. But such a generalization turns out not to be unique from a theoretical viewpoint and there are infinitely many generalizations.

So far, no established theoretical

principle leads to a simple and unique generalization.

Indeed, the original

transformations derived by Wu and M0ller are just two of the many possible generalizations. 1 In all of these cases, the spacetime of CLA frames is characterized a metric tensor of the form (W 2 , - 1 , - 1 , - 1 ) = p^ v) which may be called the Poincare metric tensor. Furthermore, because there finite transformations between inertial and CLA frames exist, the spacetime of these CLA frames is flat, i.e., they have a vanishing Riemann curvature tensor. This property is useful for formulations of field theory, including gauge theory of gravity in non-inertial frames.

297

Let us consider the transformations between an inertial frame F[(wi,Xi,yi,Zi) and a constant-linear-acceleration (CLA) frame F(w,x,y,z) which moves with a timedependent velocity p(w) along the x-axis. Suppose the metric tensors in the inertial and the CLA frames are:

=(1,-1,-1,-1),

(20.1)

P UV =(W 2 ,-1,-1,-1),

(20.2)

v

where W = W(w,r) is any real-valued function of spacetime in F(w,x,y,z) with the property W -* 1 for vanishing acceleration. Thus, the Poincare metric tensor P^v reduces to the Minkowski metric tensor TI^ in the limit of zero acceleration. Let us consider a class of transformations for an inertial frame F[(wi,Xi,yi,Zi) and a constant-linear-acceleration (CLA) frame F(w,x,y,z),2 including constant spacetime translations:

v/l =yp «oYo

«oYo

(20.3) XI = Y X + a

oYo

Y =

f^t

«oYo

yi = y+y0.

Yo = '

zj = z + z 0 ;

p = p(w) = arbitrary function,

where x£ = (w 0 , x 0 , y 0 , z 0 ) are constants and the arbitrary velocity function p(w) will be specified and discussed below. The Poincare metric tensor for the accelerated frame F(w,x,y,z) is given by (20.2) with

298

Principle of Relativity in the Physics of Accelerated Frames

W = Y2

1 jdp_ x + «- o Y o ; dw

This general transformation of spacetime is interesting because it includes the Wu transformation (19.14) and the Nteller transformation for accelerated frames as special cases. Even though p(w) is an arbitrary function of time w in the F frame, it actually has a specific and unique expression when we express it in terms of the spacetime coordinates of the inertial frame Fj

p(w)=w'+p°/a°Y°-w°.

(20.4)

Xl+l/a0Y0-X0

The fact that the right-hand-side of (20.4) does not involve any arbitrary functions implies that from the viewpoint of observers in the inertial frame F], the motion of the F frame is independent of the function p(w). (See also equation (20.17) below.) This may be interpreted as a flexibility or a 'gauge symmetry' of the time w of the accelerated frame F. When the acceleration in (20.3) approaches zero, i.e., p -» p 0 , the linearacceleration transformations (20.3) must reduce to the Poincare transformations (given w=ct and Wj = ctj),

wi = Y 0 (W + p0x) + w 0 ,

x{ =

Y 0 (X

+ P0w) + w 0 , (20.5)

yi = y + y 0 .

z! = z + z 0 ,

where P(w) -» p 0 + a 0 w, for small a 0 .

(20.6)

As long as the arbitrary function p(w) satisfies this limiting property as a 0 -+> 0, equation (20.3) will reduce correctly to the Poincare transformation (for w=ct and

Physical Properties of Spacetime in Accelerated Frames

299

W[=ct!) or to the Lorentz transformation (when x £ = 0 ) in the limit of zero acceleration. If a spacetime transformation for non-inertial frames does not have this limiting 4-dimensional symmetry, it is incomplete and probably not viable.

20b.

The singular wall and horizons

in the Wu transformation

Making the minimal departure from the Lorentz transformation and easiest non-constant choice for p(w) in (20.3), we assume p(w) = p 0 +a 0 w. The spacetime transformations (20.3) take the following form:

Wi =

Y(PO

+ «oW)

1

Po

"oYo/

«oYo

x+V

(20.7) 1 xi = Y :x+-

11

"oVo

a

oVo

yi=y.

zi=z;

where we have set w 0 = x 0 = y 0 = z 0 = 0 for simplicity, and

Y =-r=!= .

7o=-r^= >

£•>

P = P(w) = p 0 + a 0 w.

(20.8)

This Wu transformation is a minimal generalization of the Lorentz transformation obtained on the basis on limiting 4-dimensional symmetry. 1 The transformation for a differential 4-vector dx ,i =(dw,dx,dy,dz) and the explicit expression for the invariant interval ds 2 are given by (19.16)-(19.18). Note that there is a singular wall at x = -l/(a 0 Yo) in (20.7), where W or the metric tensor P0o vanish, which is unphysical. One can also verify that, on the other side of the singular wall, the differentials d wj and Wdw turn out to have different signs. The inverse transformation of (20.7) is given by

300

Principle of Relativity in the Physics ofAccelerated Frames

w = J_fw I +

p 0 /g oYo _ p J

a0 [ X ! + l / a 0 v 0

j

x = T/(XI +1 / a o V o ) 2 - ( Wl + p 0 / a o Y o ) 2

i-,

(20.9)

«oYo

y = yi,

z = zi.

As a 0 -» 0, the last terms in (20.7) and (20.9) are

singular. However, they are

cancelled by other singular terms in the Wu transformations.

Thus, the inertial

limit ct0 -» 0 is well-defined for the Wu transformations. We are used to picturing the physical world from the viewpoint of observers in an inertial frame.

Let us use the inverse Wu transformations (20.9) to

demonstrate the physical properties of the space and time axes (w,x). For constant w and xj * -1 /(a0Yo)> lines

corresponding

tne time

to

transformation in (20.9) leads to straight lines. All

constant

w

pass

through

the

same

point

(W[,X[) = (-p 0 /(a 0 Y 0 ),-l/(a 0 Y 0 )). In contrast, for constant x, the Wu transformation (20.9) leads

to

2

sets

of

hyperbolic lines

that

satisfy

the

condition

| x I + l/(a 0 Y 0 ) | > | w I + p 0 / ( a 0 v 0 ) |. The graphs in Fig. 20.1 show lines of constant x and of constant w (straight lines) in the (wi,X[) plane for the Wu transformations with zero inertial velocity, p0 = 0, and different values for constant acceleration, a 0 = -0.001,+0.001.

Fig. 20.1 Lines of constant w and x for accelerated frames with velocity fl0 = 0 and accelerations a0 = ±0.001.

Physical Properties ofSpacetime in Accelerated Frames

301

Fig. 20.2 shows the same spacetime picture with an initial velocity, p 0 =0.5, and two values of the acceleration a 0 = -0.001,+0.001.

Fig. 20.3 shows the lines with an acceleration of zero so that all lines are straight, corresponding to an inertial frame with a velocity p 0 = 0.5, as we expect from the Lorentz transformation.

Fig. 20.3 Lines of constant w and x for an inertial frame with p0 =0.5.

The graphs in Fig. 20.4 show the change of a lightcone and of the relation 2

2

s = w - x 2 , where w 2 and x 2 can be expressed in terms of w and x by using the Wu transformation (20.7).

302

Principle of Relativity in the Physics of Accelerated Frames

•1000

horizon/

•800

^ \

/

V

•4000

-

^

\

f ,u

-2000V

\

A

/

4000

X

i \ 1/

VL_—

horizon \

•4000

2000

•1CO0 i

\

/ / '^~'

•600//

\

-2000

A

i /

\ //

*^

^

V -1000 V -1600

•BOO*

'

--tooo^

20pO

/

40

\ ^

\

\ -2O00

x

Fig. 20.4 Graphs of lightcones in a frame with velocity p 0 = 0 and accelerations a 0 = +0.001 (left figure) and velocity p 0 = 0.5 and accelerations a 0 = +0.001 (right figure). It is interesting to observe that the region of spacetime consisting of all points [x + l/(a 0 Yo)l > 0 and all w from ( - l - p 0 ) / a 0 to ( l - p 0 ) / a 0 corresponds to the sector of

the

(w^Xi)

plane

between

(wj+p 0 /(a 0 Y 0 )) = -(xj+l/(a 0 Y 0 ))-

As

a

( w j + p 0 / ( a 0 y 0 ) ) = (xj+l/(a 0 Y 0 ))

o ~* °»

tnis

and

region becomes larger and

larger, eventually covering the whole (w^Xj) plane. Thus, this region of spacetime is identified with the physical world of the CLA frame F. Similarly, the region of spacetime consisting of all [x + l/(a 0 Yo)] < 0 is represented by the opposite sector with negative (X] +l/(a 0 Y 0 ). This opposite region of spacetime may be called the 'mirror spacetime' of the CLA frame. The rest of the (W[,X]) plane cannot be reached for any real w and x for a 0 * 0. The time w is physical only within the range, ( - l - p 0 ) / a 0 < w < (1 - p 0 ) / a 0 .

Outside this range, the CLA frame ceases to

exist because the velocity p(w) is greater than 1, which is unphysical. The limits of time, w -» ( ± l - p 0 ) / a 0 , in the CLA frame implies W] -» ±» in inertial frame, (everywhere in physical space except at the singular wall).

This property is

interlocked with the assumption p(w) = p 0 + a 0 w. If one makes a different assumption for the arbitrary function p(w) in (20.3), one has a different time for a CLA frame.

Physical Properties of Spacetime in Accelerated Frames

303

The Wu transformations suggest that acceleration distorts spacetime and forms a horizon that corresponds to a 'wall singularity' x = -l/(a 0 Yo)

(with

arbitrary y and z coordinates) of the coordinates for the CLA frame F or (W]+p 0 /a 0 Y 0 ) 2 = ( x ! + l / a 0 Y 0 ) 2 .

This 'singular wall' separates physical spacetime

from the 'mirror spacetime.' The location of the wall singularity depends on the sign of the acceleration a 0 and |a0Yo • The 'mirror spacetime' emerges from the quadratic equation associated with the second equation in (20.9):

/

1

\2

x+ > «oYo)

\2

l

i

\

1

=

X

{

I +

- Wr I ^° a oY0 i { "oYo J

Note that we also have a quadratic equation for the relativistic energy-momentum relation that leads to solutions with 'negative energy.'

The mirror spacetime

resembles the negative energy solution for a particle in some ways, so one may wonder whether the mirror spacetime exists in some physical sense.

Such a

question probably cannot be answered because even if it exists there is no possibility of communication between these two sectors of spacetime separated by the singular wall or the horizon. The rate of ticking of a clock and the observable speed of light in the CLA frame F near this singular wall have very peculiar properties.

Namely, as the

coordinate x approaches -l/(ct0Yo) in (20.9) (i.e., (X] +l/a 0 Y 0 ) -» (W] +p 0 /a 0 Y 0 )), the Wu transformation leads to the following results: (a) The clock stops ticking

dw'j Jdx-0

dw

304

1 _ (xI+l/a0Yo)2-(wI+po/aoY0)2 VW a0(xI+l/aoYo)3

^Q

Principle of Relativity in the Physics of Accelerated Frames

(20 10)

(b) The speed of light in the CLA frame F increases without bound

[*L] '

dW

« 0 (x 1 + l/a o V o ) 2

[H • x +1/

'ds=0

T/( I

(2an)

2

«oYo) -(Wi+|3o/a 0 Yo)

2

Result (a) shows that the rate of ticking of a clock at rest in F at (x,0,0), i.e., dx=0, slows down in comparison with a clock at (xi,0,0) in the inertial frame. Result (b) is due to the fact that the law for the propagation of light is given by ds 2 = 0 and is naturally related to (a). The ratio dw/dw t in (20.10) is positive, consistent with (X] + l / a 0 v 0 ) > 0 and (xj + l/a 0 Y 0 ) 2 > (wj + P 0 /a 0 Y 0 ) 2 > where the last relation is consistent with p 2 < 1.

20c.

Generalized

M0ller-Wu-Lee

transformation

for an accelerated

frame

Suppose one assumes the velocity function p in (20.3) for the CLA frame F(w*,x,y,z) to be

p(w*) = tanh( Yo a * w * + tanh" 1 p 0 ),

(20.12)

where w* denotes time in the accelerated frame F. The transformations in (20.3) are then the generalized M0ller-Wu-Lee transformations^

wj = x + — — sinhfa * Y0W * + tanh" 1 p„) - ^2., {

X[

a*y0)

V

J

a*

= x + —!— cosh(a*Y 0 w* + t a n r r 1 p 0 , U — , \ a * Yn) \ la*

yi = y,

z

(20.13)

i = z-

Physical Properties ofSpacetime

in Accelerated

Frames

305

When P0 = 0 , the generalized M0ller-Wu-Lee (MWL) transformation (20.13) reduces to the original Nteller transformation, 4

Wj = sinh(a * w*) x +

(20.14)

x; = cosh(a* w*)|x + — , a* J a*

yi = y.

Z] = Z .

This has been discussed by many authors. 4 '^ However, the original M0ller transformation did not involve a constant velocity p 0 and hence, did not satisfy the principle of limiting 4-dimensional symmetry.

We have generalized it so that it

involves two constants p 0 and a 0 and thus is smoothly connected to the Lorentz transformation in the limit of zero acceleration. 3 The inverse generalized MWL transformation can be derived from (20.13):

w*

—5— tantr 1 w i + p 0 / a * • tanh" 1 p 0 Xj + 1 / a *

;1/(xI+l/a*)

2

-(wI+p0/a*)2

-L «

y = yi.

306

Yo

Z = Z;.

Principle of Relativity in the Physics ofAccelerated Frames

(20.15)

\ \ w

\

4000-

4O00

3000-

3000

2000-

w

1000-

1000

0

1000 -BOO -600 -400—OT"" "

"2TJD~^6e--6blL_8dDJ_00

/ /

"TDD—-408—SB BOO 1000

X

-100D-

.,-''

2000

-1000

-2000-

-2000

-3000-

-3000

-4000 J

-4000J

Fig 20.5 Graphs of the MWL lightcone (s=0) with acceleration a* = 0.001 and velocities p o =0 (left diagram) and p o =0.5 (right diagram).

The graphs in Figs. 20.5 and 20.6 shows the changes of the relation s 2 = w 2 - x 2 for different velocities p 0 , where w 2 and x 2 can be expressed in terms of w=w* and x by using the MWL transformation (20.13).

1

w

\ ~"^.,

4000 ^

4000

3000-

3000 w

2000'

1000

1000_

100Q -800 -600 •400---20S

u

-

-

•

—

2ar-4BB—an_joo 1000

2n-4S8--soiL_805_i0(:

-1000-

/ / / /

2000'

X

-1000

-2000-

-2000

-3000-

-3000

-4000-

-4000

Fig, 20.6 MWL graph for s 2 = w 2 - x 2 with velocity (left) p 0 =0, (right) p o =0.5, acceleration a* = 0.001, s=10000.

Physical Properties ofSpacetime in Accelerated Frames

307

v/s?

600

800 1000

-1000 -

-1D00J

-0.001, (30 = 0.5.

a* = 0.001, p0 = 0.5.

= - 0 . 1 , (30 = 0.5.

a* = - 0 . 1 , /30 = 0.005.

Fig. 20.7 Diagrams showing lines of constant w and x in accelerated frames with various acceleration and velocities under the MWL transformation. The graphs in Fig. 20.7 show lines of constant w (straight lines) and of constant x in the (vfhXi) plane with various velocities p 0 and accelerations a*. These geometric properties of the accelerated spacetime are similar to those of the corresponding figures for the Wu transformations. This is not surprising because the main difference between (20.9) and (20.15) is the physical property of time:

(-l-Po),w,(l-Po)

-oo < w

< +oo.

(20.16)

In fact, based on (20.4), (20.7) and (20.15), one can show that the times w and w* are related by

308

Principle of Relativity in the Physics ofAccelerated Frames

tanh( Y o a* w* t t a n h " 1 p0) = p 0 +a 0 w = w ' + P o / ( a ° V o ) , Xi +1 /(a 0 Y 0 )

(20.17)

provided a* = a0y0. The region of physical spacetime in F(w*,x,y,z) consisting of all points [x +1 /(a * Y 0 )] > 0 with w* between -a>

and +°° corresponds to the sector of the

(wi,X]) plane between (wj +{5 0 /a*) = +(xs + l / a * ) and (wj + p 0 / a * ) = -(xj + l / a * ) . Similarly, the region of spacetime consisting of all [x + l / ( a * Y o ) ] < 0 is represented by the opposite sector with negative ( x j + l / a * ) .

This region may be called the

mirror spacetime. The rest of the (W],xi) plane cannot be reached for any real w and x. Based on the MWL transformation (20.13), as one approaches the wall singularity at x = - l / ( a * Y 0 ) , the rate of ticking of clocks and the observable speed of light in the F(w*,x,y,z) frame show the following unusual properties: (A) The rate of ticking of a clock at rest in the accelerated frame F(w*,x,y,z) at (x,0,0), i.e., dx=0, varies as a function of X] (or x and w) in comparison with a clock at (x!,0,0) in the inertial frame

Idw*) \ d w Wdx-0

-Jl-(w I + p 0 / a * ) 2 / ( x I + l / o * ) ^ a*Y 0 (x + l / a * Y 0 )

1 y0(l + a*xl)'

(20 18)

(B) Because the propagation of light is given by ds 2 = W 2 dw * 2 - dr 2 = 0, where Wx = (1 + a * Y0X) > 0, the speed of light in CLA frame F(w,x) slows down to zero,

- ^ 1 =±ll + a * Y o x | - 0 . dw *Jds-o ' '

(20.19)

Thus, there is no possibility of communication between the physical spacetime and the mirror spacetime.

These two properties in (20.18) and (20.19) are again

Physical Properties of Spacetime in Accelerated Frames

309

related. They differ from (20.10) and (20.11) because of different properties of times w and w*, as shown in (20.16) and (20.17). The contravariant 4-momentum p^1 of a particle has the same transformation properties as dx*1, which can be derived by differentiation of (20.3). Thus, we have

P?=v(Wp° + pp1),

p}= Y (p 1 + pWp°),

W=

jd^ dw' "oYo; 1

(20.20)

and pj =p , k=l,2. Following the steps from (19.30) to (19.31), one can show that the general transformation (20.3) with an arbitrary (5(w), the Wu transformation (20.7), and the generalized MWL transformation (20.13) all have a common property, namely that a particle at rest at (x,0,0) in the F frame has the same constant-linear-acceleration as measured by observers at rest relative to the inertial frame Fj because /

dpf dxj 1X

V

1 \

dp}

mdy

dwj

YfSWdw

«0m Y-

2

+aox

•• const.

(20.21)

where X stands for the general transformation (20.3) with an arbitrary velocity p(w), or the Wu transformations (20.7). We also have dp',o\

dpx\

dxi

dwi

q*m

/MWL

MWL

751 + a * X

(20.22)

-- const,

for a particle at rest in the accelerated frame at, say, a certain point (x,0,0), where x is a fixed constant. Note that (20.21) and (20.22) are the same when a* =a 0 y 0 . 20d.

Decay-length

dilations

due to particle

acceleration

Although the concept of time in the accelerated frame F differs in all of these transformations, all are smoothly connected to the Lorentz transformation in the limit of zero acceleration.

310

One might thus wonder whether any of them can be

Principle of Relativity in the Physics ofAccelerated Frames

considered physical in the sense that it can be related to some physical phenomena in an accelerated frame. The frequency of atomic radiation, the Doppler shift of the wavelength of electromagnetic radiation and the lifetime for the decay of an unstable particle are some of these phenomena. Let us concentrate on the possible effect on a particle's decay-length due to acceleration. & A simple setup to perform such experiment is to create a potential drop over a distance L, say from x a to X12 = x^ + L. One could place detectors at x n and X]2 to count the number of particles (charged pions for example) in the beam before and after traveling a length L. There is an additional complication however, i.e., the spacetime transformations we have been discussing have a metric tensor P 00 = W 2 with a non-trivial space-dependence. One can usually neglect this nuisance property by setting x=0, i.e., assuming that the particle is located at the origin of the accelerated coordinate system. However here, we want to examine this x-dependence explicitly and graphically for a particle located at any point in the CLA frame. Because quantum field theory has not yet been formulated for accelerated frames, we shall make some reasonable assumptions. For example, we will assume that the usual exponential decay-law holds in accelerated frames, N = Noexp[-Aw/w0],

Aw = w 2 - w 1 ,

(20.23)

where w 0 is the decay lifetime of a particle at rest in an accelerated frame. We shall assume that w 0 is approximately the same as the lifetime that has been measured in (inertial) laboratories. This approximation should be good when the acceleration is small. (For further discussions, see section 20e below.) Based on this law, we shall now deduce and examine the predictions for the lifetime dilation of an unstable particle in flight that is being accelerated by a potential field. We use the inverse Wu transformation (20.9) to express w 2 - wi in terms of spatial coordinates in the inertial laboratory frame. We have

Physical Properties of Spacetime in Accelerated Frames

311

W2 - Wi =

W I2 + PQ /(a 0 Yo) x I 2 +1 / ( a 0 y 0 )

/

w a + p 0 /(a 0 Yo) x n +1 /(a 0 Y 0 )

2 X2+l/(a 0 Yo) X I2 + l / ( a 0 Y o ) /

Xi+l/(a0Yo) Xa+l/(a0Y0)

(20.24)

where the particles are at rest at x = -x 2 = xj in the CLA frame. Therefore, if there are N 0 particles at x n , then the number of particles remaining at x !2 = XQ + L is given by (20.23) and (20.24)

N(xI1+L) = N 0 (x I1 )exp

-1 a0w0

X 2 + l / ( a 0 Y o2x\ )

X I2 +l/(a 0 Y 0 )J

]

X1+l/(a0Yo) Xn +1 /(a 0 Y 0 )

(20.25)

where x = X2 = Xj is fixed. This is the formula for the number of unstable particles remaining after starting from an arbitrary point X] and traveling a distance L with a constant-linear-acceleration. Note that the exponential part in (20.25) is dependent on the coordinate variable x or x I; where these two variables are related through the transformation (20.9). In the limit of zero acceleration, a 0 -»0, the formula reduces to the usual formula in special relativity

N(x I1 +L) = N 0 (x I1 )exp

Po w o

(20.26)

for particle decay in flight with the velocity p 0 . Note that the exponential part in (20.26) is now independent of the coordinate variables x and Xj, as it should be for inertial frames.

312

Principle of Relativity in the Physics of Accelerated Frames

On the other hand, if one uses the generalized MWL transformation (20.15) to express the law (20.24) with w -* w*, one has the result:

N(xn+L) = N 0 (x I 1 )exp| a*Y 0 w 0

T-tanh-1Jl-[X

+1/(a 1fo) * l -tanh-1 X12 + l / a * j I

x

fx + l/(a* Y o )^ X.ll+1/a*

(20.27)

j

Figs. 20.8 and 20.9 show graphically the different predictions of the two formulas. Both of them are insensitive to the coordinate x in (20.25) and (20.27).

R I WU 0.1776-

0.1774-

0.1772-

0.177-

special relativity 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

X Fig. 20.8 Coordinate dependence of the concentration R = N / N 0 . Comparing Wu transformation and special relativity p 0 = 0.5,a 0 = 0.001,L = 7.8, w 0 = 7.8.

Physical Properties ofSpacetime in Accelerated Frames

313

R

0.36-

MWL

0.340.32: 0.30.28 0.260.240.22 0.2i

WU

0.180

speniai relativity' 0.2 0.4 0.6 0.8

1

1.2

1.4

1.6

1.8

X

Fig. 20.9

Coordinate dependence of the concentration R = N / N 0 . Comparing

MWL and Wu transformations p 0 = 0.5, a 0 = 0.001, L=7.8, w 0 = 7 . 8 . 20e.

Discussion

Although the

velocity function

p(w)

in

(20.3)

is arbitrary,

the

transformations for the differentials dx11 are as simple as that in (19.16) (where p(w) = p 0 +a 0 w). From (20.3) with arbitrary velocity p(w), we obtain

dw] = v(Wdw + pdx),

dX[ =v(dx + pWdw),

where W = Y2(x + l/a 0 Yo)dp/dw.

dV] = dy,

dz[ = dz,

(20.28)

As usual, the set of the generalized Lorentz

transformations (20.7) form the Wu group that includes the Lorentz group as a limiting case of zero acceleration. 1 However, in order to see the group properties one must consider, for example, the Wu transformation between two CLA frames F(w,x,y,z) and F'(w',x',y',z'),

314

Principle of Relativity in the Physics ofAccelerated Frames

Y(PO+«0W)

a0Yo

Po «oYo

:

P'o

l ' ( P » ' ) V

«oTo (20.29)

"oYo

a 0 Yo

y = y,

x+ I a 'oY'S/

z=z;

rather than just (20.7) for an inertial and a CLA frame. After some calculations, one can express the unprimed variables in (20.29) in terms of primed variables and constants. Similarly, the set of transformations (20.13) also form a group. These symmetry group properties are important for the formulation of field theories in accelerated frames. To test the two decay laws (20.25) and (20.27) involving the two constant accelerations a 0 and a*, it is reasonable to compare them with the relation a

* = a oYo because their corresponding times w* and w are related by equation

(20.17). If there is enough data, we expect that the formula (20.25), if it is correct, will enable us to determine experimentally the 'rest lifetime' w 0 of a particle in an accelerated frame. Experimental tests of the sole a 0 -dependent effect in the decay formulas is particularly interesting because it is dependent purely on the acceleration rather than on the velocity p(w)=p 0 +a 0 w.

In discussions of accelerations, physicists

usually make the simplifying assumption that any effects of acceleration appear only through the velocity p(w) = p 0 + a 0 w).

However, strictly speaking, this

assumption appears to be inconsistent with the principle of limiting 4-dimensional symmetry, which lead to a distinct a 0 -dependence in the decay formulas (20.25) and (20.27).

To see this a 0 -dependence in a simple form, we set x = xj = 0 in

(20.27) and obtain

N(L) No(0)

•• e x p

a0W0

•Po

(20.30)

(1 + a0Y0L)

Physical Properties ofSpacetime in Accelerated Frames

315

0.4- R

0.3-

X

0.2-

/ 0.1-

S

/ -i

0.02

0.04

•

«

1—

0.06

0.08

0.1

Fig. 20.10 R=N(L)/N o (0) in respect to a 0 (see eq. (20.30)) (L=7.8, w 0 = 7.8, p 0 =0.1, vertical line at a o =0.063).

1 R 0.8

^

0.6

0.4

0.2 i 0

0.02

0.04

0.06

0.08

0.1

Fig. 20.11 R=N(L)/N o (0) in respect to a 0 (see eq. (20.30)) (L=7.8 w 0 Po = 0.943, vertical line at a o =0.0025).

316

Principle of Relativity in the Physics of Accelerated Frames

Suppose a charged pion enters a potential gradient of 10 MeV per meter, AV= 10 MeV/m, with an initial velocity p 0 . Its acceleration can be calculated using (20.21). One obtains <xo=0.063/m for (50=0.1, and a o =0.0025/m for p 0 = 0.943. The a 0 -dependent behavior in (20.30) is shown in Figures 20.10 and 20.11. When p o =0.1, we use the values a o =0.030/m, 0.063/m, 0.09/m. And when p o =0.943, we use the values a 0 = 0.0012/m, 0.0025/m,

0.005/m. 6

In quantum field theory, the decay rate for accelerated unstable particles is formally the same as in (17.72) with a time-dependent factorr((5)=r 0 (l-p 2 ) 1 / 2 . Making a natural generalization of the usual exponential decay law by assuming that its differential form, dN/dw=-rN is also valid in an inertial frame with a timedependent decay rate r, the decay law for particles at rest in a CLA frame is then

N / N 0 = e x p [ - r o / f H 3 2 d w , ] = exp -r 0 Ajsinh l

A

-sinh

where we have used (18.9), P=(wi+p 0 /a 0 Y 0 )/(xi+l/a O Yo)> A=x+l/a 0 Yo

(20.31)

an

d B^ =

w I k + p 0 /a 0 Yo. k=l,2. On the other hand, from the point of view of limiting Lorentz and Poincare invariance, the differential form of the invariant decay law dN/ds=-r 0 N should hold not only in all inertial frames, but also in non-inertial frames in the limit where the acceleration goes to zero. If we make the additional assumption that this differential form is valid in non-inertial frames not only in the limit of zero acceleration but also for non-zero accelerations, then we obtain the same result for N/N 0 as the one based on the field theory approach. As expected, this expression for N/N 0 reduces to the usual decay law for inertial frames in the limit of zero acceleration.

Thus, the results in equations (20.23) and (20.24) are valid only for

CLA frames in the limit of small accelerations. 7 It is important to determine experimentally the correct physical time in accelerated frames before attempting to formulate physical theories in accelerated frames. Thus, experiments that can determine the correct physical time in accelerated frames are crucial to our understanding of the physics of non-inertial frames.

Physical Properties ofSpacetime in Accelerated Frames

317

1.

Jong-Ping Hsu and Leonardo Hsu, Nuovo Cimento B 112, 575 (1997) and Chin. J. Phys. 35, 407 (1997). Jong-Ping Hsu, in Symposium on the Frontiers of physics at Millennium (Edited by Yue-Liang Wu and Jong-Ping Hsu, World Scientific, Singapore, 2001) pp. 321-328.

2.

D. T. Schmitt and J. P. Hsu, Inter. J. Mod. Phys. A 20, 5989 (2005).

3.

J. P. Hsu and S. M. Kleff, Chin. J. Phys. 3 6 768 (1998); in JingShin Theoretical Physics Symposium

in Honor of Professor Ta-You Wu (Ed. J. P.

Hsu and L. Hsu, World Scientific, Singapore, 1998), pp. 348-352. 4.

C. M0ller, Danske Vid. Sel. Mat.-Fys. 20, No. 19 (1943); see also The Theory of Relativity (Oxford university press, 1952), Chapter VII; Ta-You Wu and Y. C. Lee, Intern. J. Theoretical Phys. 5, 307 (1972). Ta-You Wu, Theoretical Physics, vol.4, Theory of Relativity (Lian Jing Publishing Co., Taipei, 1978) pp. 172-175.

5.

T. Fulton and F. Rohrlich, Ann. Phys. 9, 499 (1960); T. Fulton, F. Rohrlich and L. Witten, Nuovo Cimento XXVI, 652 (1962); E. A. Desloge and R. J. Philpott, Am. J. Phys. 55, 252 (1987).

6.

The constant acceleration a 0 acceleration a 0

(in unit 1/m)

is related to the usual

2

(in unit m/s ) by a 0 = a 0 / ( Y 0 c 2 ) , as one can see by

comparing equations (18.6) and (18.8). We can estimate that a o =0.005/m corresponds to an extremely large acceleration a o ~ 1 0 1 3 m / s 2 for y 0 ~l7.

The decay law in a CLA frame differs from (20.31). Suppose particles are at rest in a CLA frame.

Based on the Wu transformation (19.14), we obtain

(N/N0)cLA=exp(ZWu),

Zwu=-r o ;wdw = -r 0 (x + l / a o Y ^ [ t a n h - 1 p 2 - t a n h - 1 p 1 ] , l

+a

w

where P k = P o o k > k=l,2. We have used the decay law dN/ds=-r 0 N. On the other hand, if one uses the generalized MWL transformation (19.7), one 2

has (N/N0)cLA=exp(ZMWL). where Z M wL=-roJW x dw* = -r 0 (l + a*YoX)[w 2 -w 1 ]. l

318

Principle of Relativity in the Physics ofAccelerated Frames

21. Extended Lorentz Transformations for Accelerated Frames and a Resolution to the "Two-Spaceship Paradox" 21a.

The two-spaceship

paradox

For years, the "two-spaceship paradox," including the effect of a constant linear acceleration has been discussed in the literature-'- based on special relativity (with d 2 s = c 2 dt 2 - d r 2 for an inertial frame) and d 2 s = (l + a 0 x / c 2 ) 2 c 2 d t 2 - d r 2 = ds 2 for a non-inertial frame with a constant linear acceleration a 0 .

However, the

discussions have been based on intuitive arguments rather than an explicit spacetime transformation between the two frames, one of which is accelerated. In chapters 16 and 17, we obtained a spacetime transformation between an inertial frame Fi(wi,X],yi,Zi) and a frame undergoing constant linear acceleration (CLA) frame F(w,x,y,z) as an extension of the Lorentz transformation, which is also characterized by a metric with the form d 2 s = W 2 dw 2 - dx 2 - dy 2 - dz 2 .

In this

chapter, we use the transformation to resolve the apparent paradox. The essential parts of the 'two-spaceship paradox' are as follows: 1 (a)

Consider two identical spaceships, A and B, which at first are at rest in an

inertial frame Fj, a distance L apart. At time w I = ctj=0 the spaceships accelerate in the same direction parallel to the line that joins them. As viewed from Fj, they undergo the same acceleration for the same duration, then stop accelerating at the same time after reaching a steady speed u. (b) The apparent paradox arises when asking the question "What is the distance between the two spaceships after they reach the steady speed u, observed from Fi?" Arguments can be advanced either for the distance to remain unchanged (because both spaceships have undergone the same motion) or for the distance between the

319

ships to be Lorentz contracted L' (=L-\/l-(u/c) 2 ) (treating the spaceships as the two ends of a long rod in motion). Although the dynamics of

a point particle undergoing a

constant

acceleration can be treated in the framework of special relativity, the constantlinear-acceleration of a reference frame is, strictly speaking, beyond special relativity because of the extended nature of the reference frame.

In order to

discuss the two-spaceship problem, we must have transformations that include two parameters, the relative instantaneous velocity and the acceleration of the noninertial frame. As we shall see, such transformations for accelerated frames (with a metric of the form, d 2 s = (1 + a 0 x / c 2 ) 2 c 2 d t 2 - dr 2 s ds 2 imply that if the engines of spaceships A and B are turned off at the same time w n = W12 as measured in the inertial frame Fj, then their final velocities relative to Fi cannot be the same, as was assumed in the statement of the paradox. Surprisingly, this peculiar property (see eq. (21.14) below) is not in conflict with experimental results and is consistent with accelerated Galilean transformations at small accelerations and velocities. It is a natural result of the limiting four-dimensional symmetry of accelerated frames. As discussed in chapters 16 and 17, one can find two transformations relating the coordinates of an inertial and a non-inertial frame with constant linear acceleration, the Nfoller and Wu transformations.2.3 However, in discussing the twospaceships paradox, neither is ideal. The Nteller transformation involves only the acceleration of the non-inertial frame as a parameter and reduces to the identity transformation rather than the Lorentz transformation, when the acceleration is zero. ^ The Wu transformation involves both the acceleration of the non-inertial frame a 0 and the relative instantaneous velocity of the two frames |30 at time w=0. However, when the acceleration is zero, it reduces to the Lorentz transformation with a relative velocity p 0 between the two inertial frames whose origins coincide at w = 0. Thus, it is inconvenient to use the Wu transformation when discussing a situation such as the "two-spaceship paradox" because the acceleration of a frame (the frame of the two spaceships) becomes zero at a time other than w = 0. To work around this problem, we first use generalized forms of both the Nteller and Wu transformations, both of which smoothly reduce to the Lorentz

320

Principle of Relativity in the Physics of Accelerated Frames

transformation when the acceleration a 0 is zero. The generalized forms of these transformations can be used to obtain the Lorentz transformation between the inertial frame Fj and the inertial frame occupied by the two spaceships after they stop accelerating without any additional adjustments.

Such transformations for

accelerated frames are necessary in order to discuss operationally any changes of lengths that are related to constant linear accelerations. The generalized M0ller and Wu transformations are particularly suited for the clarification of the "twospaceship paradox" because they enable us to discuss length contractions from the viewpoint of both inertial observers and the spaceship pilots before, during and after the acceleration.

21b.

Generalized

Nteller and Wu transformations

Let us consider first the Wu transformations.

In the following discussion, we

assume that both the velocity and the acceleration are along the x-axis, ignoring the unimportant y and z axes. The general Wu spacetime transformation between an inertial frame F[(wi,X]) and a CLA frame F(w,x) can be written as^

cti=vP

'x + 1 '

Xi=Y X +

1

+ WA,

(21.1)

\ + *A,

where

Y =- 7 —

.Jl-P 2

.

Ya=-j

=

'

P = « o ( w - w a ) + pa,

(21.2)

VI "Pi

WA=VaWa--^-, «oVa

XA=YapaWa

a

.

(21.3)

oYa

Extended Lorentz Transformations for Accelerated Frames

321

The quantity p a is the velocity at a given time w a , p a = Po + c t o w a- If

one

chooses

w a = 0, then p a is the initial velocity p 0 . In this special case, (21.1) becomes the Wu transformations discussed in section 19b. From the relation for p in (21.2), we see that the limit a 0 -» 0 is the same as w -» w a . Therefore, the limit a 0 -» 0 is always understood as taking the time w to be w = w a , so that we have p = p a in this limit. The general Wu transformation (1) can be derived on the basis of limiting 4dimensional symmetry together with the linear function p = p(w) in (21.2).3 preserves the invariant interval d2s = d w 2 - d x 2 = W 2 d w 2 - d x 2 ,

(21.4)

W2=v4(Ya2+«0x)2.

<21-5)

If one introduces another time variable w* given by

w * = —!— ( t a n h - 1 [ P a + a 0 ( w - W a ) ] - t a n h - 1 [ P a - a o W a ] ) , «oYa V '

(21.6)

the metric in (4) can be written as

d2s = W 2 d w * 2 - dx2,

W M = ( 1 + Y2a0x),

(21.7)

where the metric tensor goo = W^ is time-independent, and

^YaVdw,

(21.8)

so that the metric ds 2 in the CLA frame F(w*,x) can be written as

ds 2 =(1 + Yaa 0 x) 2 dw* 2 - dx 2 .

322

Principle of Relativity in the Physics of Accelerated Frames

(21.9)

It

The spacetime transformation (21.1) can then be expressed in terms of (w*,x,):

sinh|Y|a 0 w* + tanh 1 P O ) + W A

W[ «oYa

(21.10) x

r

cosh(Yaa0w * + tanh

x+-

1

p 0 ) + XA

"oYa

where P 0 = P a - a o w a -

This is the generalized M0ller transformations.

One can

verify that both the generalized M0ller transformation (21.10) and the Wu transformation (21.1) include the Lorentz transformations as the limiting case when cc0 -* 0 at any time w a of the CLA frame F:

W] =Ya( w + P a x l

x

i =Ya( x + Pa w )-

The times in these transformations (21.1) and (21.10) can be operationalized by computerized clocks as described previously in chapter 18.4 Our results for the two different times w and w* can be summarized as follows: (A') When w a = 0 (i.e., p a = p 0 ), the transformation (21.1) corresponds to the usual Wu transformations (19.14), characterized by a linear relation (21.11)

P = P0 + a 0 W

and

a finite range for the time variable w in

the CLA frame F(w,x),

( - l - P 0 ) / a 0 < w < ( l - p 0 ) / a 0 . Note that the maximum value w m a x = (l-p 0 )/cc 0 for the time w in the CLA frame F corresponds to p = 1 and, hence, wj = +°° for the time in the inertial frame F^WJ.XJ). The inverse transformation to (21.1) is:

Wi-Wi

w =

1

X

X

Pa

—

«O( I- A)

J a

- - + wa

(21.12)

"O

Extended Lorentz Transformations for Accelerated Frames

323

X = T/(XI-XA)2-(WI-WA)2

l

—r.

a 0 7a

(B') When pa = 0 and wa = 0 (or p0 = 0), the transformation (21.10) reduces to the usual M0ller transformation, i.e.,

Wj = x + —sinh(a*w*), and yi = y,

X] = x + —cosh(a*w")

,

Zj = z. The Ntoller transformation is characterized by

(i) a non-linear relation between velocity p and time w* , p = tanh(a0VaW*+tanh-1p0) = tanh(a0VaW*), and

(21.13)

(ii) an infinite range for the time variable w* of the CLA frame F(w*,x), -» < w* < +°°. The inverse of the generalized M0ller transformations can be derived from equation (21.10). 21c.

Motion and length contraction

involving

accelerations

Let us now consider the linearly accelerated motion of an arbitrary point x, i.e., x=k=constant or dx=0, as measured in an inertial frame F[,

N dw

l lix=k

W; - W A . Xi - X A x=k

W; - W A

"J l

(21.14)

(x + l / a 0 Y ^ ) 2 + ( w I - W A ) 2 v„W

where we have used (21.1) and (21.12). This result holds for both the generalized Wu and Nfeller transformations, implying that at a given time Wj, the velocity dx[ /dw] of a particle at (ki,0,0) is different from the velocity of a particle at point (k2,0,0), as measured by an observer in an inertial frame F[.

This result

substantiates our previous observation concerning the final velocities of the two

324

Principle of Relativity in the Physics of Accelerated Frames

spaceships in section 1. Note that when one compares the velocities of two objects at the same time wj in the inertial frame, one should not set a 0 - » 0 because this limit corresponds, by definition, to events that occur at the same time w = w a in the CLA frame F. Suppose the two spaceships are at rest in the CLA frame F along the x-axis. What is the distance the two points occupied by the spaceships x^ and x 2 ? Since there is a wall singularity at x s = -l/(a 0 Ya) in the extended Lorentz transformations (21.1), (21.10) and (21.12),2.3 the values of Xj and x 2 must both be larger (more positive) than x s in order for them to be on the physically occupiable side of the wall singularity.

The left-hand-side of the singular wall is unphysical because it

cannot be reached by any physical signal from the origin of a CLA frame. However, there is no restriction on the distance between these two points.

If the distance

between these two points is measured by two inertial observers at the same time, W

I2 = WU = w i o > t n e inverse transformations (21.12) lead to

(x2-Xi)ai=(xI2-xA)Jl^2"_(Xll-xA)jrp[,

p

.

=

^ W A

;

.

(21.15)

= U ;

*Ii - XA

where the subscript QI denotes the condition w !2 = Wu = W]0. To the first order in ct0, we have the following approximation:

x

2 - xi)

-

Ya (xi2

- xn)

»o7a

-2Y a w a Pa(xi2 - *ll) + ^j2 "

- ^ [ - 2 P a W i o ( x i 2 - x a ) + x2 2 -x2] .

Extended Lorentz Transformations for Accelerated Frames

x

n

(21.16)

325

On the other hand, suppose the two corresponding distances x !2 - x^ and x 2 - xj are measured at the same time w 2 = wi = w a of the CLA frame F. The generalized Wu transformations (21.1) lead the following exact result:

[xi2-Xn]

=Ya(x2-Xi),

Ya = -f . Vi-CPo + aoWa)2

(21.17)

where the subscript Q. denotes the condition w 2 = Wi = w a . The generalized M0ller transformation (21.10) then yields

[ x i 2 - x n ] a i = ( x 2 - x 1 ) c o s h ^ a o W * + tanh- 1 p 0 j = Y a ( x 2 - x 1 ) ,

(21.18)

where the subscript 0_* denotes the condition w * 2 = w *j = w * a , and we have used the relation (21.6) for times w* and w. Thus both the generalized M0ller and Wu transformations of spacetime lead to exactly the same result (21.17) or (21.18) which depends only on the velocity p a at the time w a of measurement. This result is similar to that in special relativity, i.e., roughly speaking, a "moving" meter stick (at rest in Fj) appears to be shorter, as measured by observers in the CLA frame F, as shown in (21.17) and (21.18).

2 Id.

Discussion

In the previous argument, it does not matter whether the distance (x2 -Xj] in (21.15) is the length of a spaceship or the distance between two spaceships A and B that are at rest in the CLA frame F. During acceleration, both the distances between the two spaceships and the lengths of the spaceships undergo the usual Lorentz contraction in addition to a change due to the constant acceleration. If both the initial velocity and the acceleration are zero at time w a = 0, i.e. p a =p 0 = 0 , and a 0 = 0, then equation (16) reduces to x 2 - xj = Xj2 - x^ for w I2 - w fl = 0. Thus, there is no length contraction before acceleration.

326

After accelerating to reach a

Principle of Relativity in the Physics of Accelerated Frames

constant velocity p a , i.e. setting a o = 0 at time w a in (16), both the distances between two spaceships and the length of a spaceship are contracted by just the usual Lorentz contraction, Tx2 -Xi]

= Ya(xi2 _ x u ) -

A qualitative picture of a reference frame F with a constant-linearacceleration is as follows: The spacetime of accelerated frame is non-uniform, in sharp contrast to the uniform spacetime of inertial frame. An inertial frame can be pictured as an (idealized) rigid scaffolding, while a CLA frame F corresponds to scaffolding with a non-uniform deformation. Along the x-axis, the Moller and Wu transformations map only the section of spacetime between the singular wall and x =+oo in an accelerated frame F to the entire spacetime extending from x = - » to x = +t» in an inertial frame. The limits on the time variable w -» [(±l-(3 a )/a 0 + w a ] in the CLA frame implies wj -»±oo in the inertial frame everywhere in physical space except at the "singular wall,' x = -l/(a 0 Y a ).

It is interesting that as one approaches

this 'singular wall' in the accelerated frame the clocks slow down relative to the nearby clocks in an inertial frame F[, and the speed of light increases without limit. At the singular wall, the clocks stop completely and the speed of light becomes infinity.^

These

properties are

interlocked with

the

assumed

relation

p(w) = p a +a 0 (w - w a ) for the Wu transformation (21.1). In general, intuitive arguments related to acceleration are not reliable because of the non-trivial physical properties of accelerated frames, as shown in (21.1) and (21.10). Once we postulate the metric d 2 s = (l + a 0 x / c 2 ) 2 c 2 d t 2 - d r 2 for a constant-linear-acceleration (CLA) frame F(w,x), then a particle at rest in this CLA frame does not satisfy d2X[ / dw 2 = constant, as measured by observers in an inertial frame Fj. This is because the velocity of a particle moving with d2X] / d w 2 = constant will become larger than the speed of light for sufficiently large wj, leading to unphysical results.

As argued previously, the only relativistically

consistent meaning for "constant-linear-acceleration" is one in which the relativistic momentum p I x (or energy p I 0 ) of a particle undergoing CLA satisfies dpi x /dwj= constant (or dp I 0 /dX[=constant), as measured in an inertial frame. 3 For example,

Extended Lorentz Transformations for Accelerated Frames

327

particles in a high energy linear accelerator that has a constant potential drop per unit length along the particle beam have a constant linear acceleration. Thus, the generalized M0ller and Wu transformations can be used to discuss the two-spaceship problem involving a constant-linear-acceleration.

Using these

specific transfomations gives the measurements of length and time well-defined operational meanings and hence, there is no "two-spaceship paradox."

328

Principle of Relativity in the Physics of Accelerated Frames

1.

For earlier discussions related to this paradox, see E. Dewan and M. Beran, Am. J. Phys. 27, 517 (1959); A. A. Evett, Am. J. Phys. 40, 1170 (1972); J. S. Bell, Progress in Scientific Culture, Vol. 1, No. 2, summer 1976; H. F. Yau, Am. J. Phys. 50, 278 (1982); R. d'E Atkinson, Am. J. Phys. 48, 581 (1980). For a recent discussion, see T. Matsuda and A. Kinoshita, Association of Asia Pacific Physical Societies Bulletin, Feb. 2004, pp. 3-7.

2.

C. M0ller, Danske Vid. Sel. Mat.-Fys. 20, No. 19 (1943); see also The Theory of Relativity, (Oxford University Press, 1952), Chapter VII. Ta-You Wu and Y. C. Lee, Intern. J. Theoretical Phys. 5, 307 (1972). Ta-You Wu, Theory of Relativity (Lian Jing Publishing Co., Taipei, 1978) pp. 172-175. J. P. Hsu, Nuovo Cimento B 108, 949 (1993), and B 109, 645 (1994).

3.

Limiting 4-dimensional symmetry means that any accelerated transformation of spacetime must reduce to the Lorentz transformation (that displays the 4dimensional symmetry) in the limit of zero acceleration. J. P. Hsu, Einstein's Relativity and Beyond-New Symmetry

Approaches

(World Scientific, 2000)

Ch. 21, and J. P. Hsu and Silvia M. Cleff, Chin. J. Phys. 36, 768 (1998). 4.

One can set up a grid of clocks in an inertial frame by the usual procedure. The clock system in CLA frame F can be realized as follows: Suppose a computer clock can accept information concerning its position in the Fj frame, obtain Wj from the nearest F[ clock, and then compute and display w [or w*] using the inverse transformations (21.11) [or (21.12)]. See J. P. Hsu in ref. 2. The problem of physical time for the CLA frame is still open. It is reasonable to assume that the physical time for the CLA frame should be able to describe the lifetime dilatation of particles decay in flight (and the Doppler effect of a moving source) with constant-linear-acceleration.

This

can be determined by experiments in the future. 5.

D. T. Schmitt and J. P. Hsu, Intl. J. Mod. Phys. A 20, 5989 (2005).

6.

If one uses another time variable w* in (6) for the accelerated frame F, then the speed of light slows down to zero at the singular wall.

Extended Lorentz Transformations for Accelerated Frames

329

22. Dynamics of Classical and Quantum Particles in Constant-Linear-Acceleration Frames 22a. Classical frames

electrodynamics

in

constant-linear-acceleration

Within the four-dimensional symmetry framework of taiji relativity with the usual basic units (length, time, mass), quantum electrodynamics has only two universal and fundamental constants, J = 3.5177293xl0_38g-cm and e = -1.6021891xro- 20 (4jt) 1/2 (g-cm) 1/2 , instead of the usual three, c, h, e (in the electrostatic unit, esu). 1 However, based on physical considerations, we have also argued in chapter 12 that from the most fundamental point of view, in which there is only one unit of measurement, quantum electrodynamics has only one truly universal and fundamental constant e = e = V4n /V137.036. In this chapter, we shall show that these results also hold in the present formalism of physics in accelerated frames, in which the speed of light is not a constant. Let us assume that accelerated frames satisfy the Wu transformations in (19.14). Since the speed of light in an accelerated frame F is no longer a universal constant, we assume that the invariant action for a charged particle moving in the electromagnetic potential a^(x) can be written as 2 S= ("(-rnds-ea^dx^-i ff^vf^vWd4x

= jLdw-ij*f ( i v fi i V Wd 4 x,

ds 2 =W 2 dw 2 -dx 2 -dy 2 -dz 2 , f|iv=Vv-avV

dx^=(dw,dr),

(22.1)

^-detg^ =W = Y2(Y82+a0x)> g^P^,

330

(22.2)

(22.3)

dX(i = g^ v dx v =(W 2 dw,-dr),

g|lv

=(W 2 ,-1,-1,-1),

in a reference frame F that moves with a constant-linear-acceleration (CLA). The differential of the coordinate dxfe(dw,dr) is, by definition, a contravariant vector. The covariant coordinate dx^is given by (22.3). Note that the i n v a r i a n t action S for a CLA frame does not involve the constant c and that Wd1^ = yj-det

g^

xdwdxdydz is the invariant volume element in taiji spacetime. We use taiji-time w as the evolution variable, so that the Lagrangian L, defined in (22.1), takes the form, L = -m-^W 2 - p2 _ p 2 _ p 2 _ g ( a Q

+ aipi)f

i = :L|2,3,

(22.4)

p = dr/dw = ( p ^ . p 3 ) - (px,Py,Pz). Note that p in (22.4) is the 'velocity' of the particle measured in terms of taiji time w.

In the limit a 0 -» 0, F is an inertial frame and, hence e

and a^

correspond to the charge e(in esu) and the usual electromagnetic potential A^(ct,r) by e = e/c and a^fw^) •» A^(ct,r)/c respectively.

The canonical

momentum Pj of a particle in the CLA frame F is defined by

Pi=-P1,

P i = - - ^ - = Pi+eai,

-mrp x -mrp y -mrp z W

r°

'

W

, ] . , 2 w2 ' \ / l -0p z //W ^

^

'

W

i = 1,2,3;

k

ikP .

(22.5)

P x - P ^ - P i , etc.,

1 P 2 =p£+p£+p|—PlP . ry

""

''"

'"

'""" '

'

P'=^dw

(22.6)

(22.7)

The "Hamiltonian" H = P 0 , which has the same dimension as that of P;, is defined by

Dynamics of Classical and Quantum Particles

331

2 2 Pn = ' i L p l _ L = Po +eag = \(Pi - e a j ) +m +eao =H ;

(22.8)

lap' „,,, o p 0 = mrW = g0oP .

o P =

m

dx° dw - r - =m - i - ds ds

,.._ „, ( 22 -9)

Note that the contravariant momentum p^ and the covariant momentum p^ are related by p ^ g n v P ^ , i.e., Pn=(po,Pi,P2,P3)=(W 2 p 0 ,-p 1 ,-p 2 ,-p 3 )3(W 2 p 0 ,-p) where the function W is given in (22.2). We observe that the covariant momentum Pn=(P0iPi>P2.P3) in (22.6) and (22.9) can be written as p^=mdx^/ds=mg^ v dx v /ds. Since m and ds are invariant, p^ should transform as g(xvdxv- From the transformations of dx v in (19.16) and g(iv given in (19.18), we obtain the transformation of the covariant momentum

Pio-Y^-PPi

,

Pn-YPi-P^-l.

Pi2=P2,

Pi3=P3;

(22.10)

B = a w + p 0 = ( w I + Po / a oYo) . -\|(xi+l/v 0 a 0 ) 2 -(x + l / a 0 v 2 ) 2 ^ (X!+l/a 0 Y 0 ) (xi+l/a 0 Y 0 ) where we have used dx^=g (iv dx v , dxj =n dxj v , r\ =(1,-1,-1,-1) and (19.15). Note that p in (22.10) is the velocity of the CLA frame F and differs from p i n (22.4)-(22.9), where p= (p 1 ^ 2 ^ 3 ) • (Px,Py,Pz) is the velocity of a particle. This transformation (22.10) allows us to see how the particle's energy p I0 increases as a function of distance it travels in an inertial laboratory.

Suppose

the particle is in the CLA frame at x'=(x o ,0,0)=constant, so that dxVdw=0, r=l and p0=mW in (22.7) and (22.9). We have p I0 =mY=m/Vl-P2, which leads to

ma ^PI0= ° - constant, dx I (T^+a0x0)

332

Principle of Relativity in the Physics ofAccelerated Frames

(22.11)

where we have used p in (22.10) with x=x 0 =constant.

This result gives the

operational meaning to the concept of constant acceleration in an inertial laboratory such as the Stanford Linear Accelerator Center. To be more specific, if xo=0 and the initial velocity Po=0, the constant acceleration a 0 of a charged particle with mass m can be obtained by measuring (l/m)(dpio/dxi), w h e r e (dpio/dxi) is related to the potential gradient of the accelerator. The Lagrange equation of motion of a charged particle can be derived from the invariant action (22.1). We obtain

mDlV m —— = ds v

uv =

dx v

eV V ' ,

e=e = V w - \ / l 3 7 . 0 3 6 .

Du

(22.12)

^=VvdxVsDvVxV'

ds dU

DyU^ = — ^ - rP^Up * dyu^ - rP^vUp,

DyuM- = dyuM- + r^vpuP

TPnv = - g p a ( V g a v + 3 v g a n - dag|iv) = I^Vf*.

where ;v or Dy denotes the partial covariant differentiation 3 with respect to x v . The Christoffel symbols rP^v can be calculated with the metric tensor g^v given by (22.3) for the space of a CLA frame with the Wu transformation. On the other hand, if one works in the space of a frame with the generalized MWL transformation, one uses the metric tensor in (19.8), i.e., g *(j,v=(W x 2 ,-l,-l,-l) where Wx=l+Yo°<*x, to calculate the Christoffel symbols and so on. For a continuous charge distribution in space, the invariant action for the electromagnetic fields and their interaction is assumed to be

Sem = - / a ^ f W d 4 x - ± J \ v f ^ W d 4 x ,

(22.13)

which leads to the following Maxwell's equations

Dynamics of Classical and Quantum Particles

333

ffiv —D|xav -D-ya^ —fl^av- dva^ , Dafjiv

=

^afixv _rf>nafpv - rPvaf^p.

(22.14)

These Maxwell's equations are invariant under the Wu transformations for CLA frames. Note that the coordinates in the Wu transformations (19.14) for a CLA frame F(w,x,y,z) is not arbitrary, in contrast to the coordinate transformations in general relativity.

Rather, they are a particular set of coordinates with a

metric tensor given by (19.18) and with a vanishing curvature tensor.

Riemann-Christoffel

Since the Wu transformations is smoothly connected to the

Lorentz transformations in the limit of zero acceleration, the coordinates of a CLA frame have the same physical significance as that of an inertial frame. 2 2 b . Quantum particles and Dirac's equation in a CLA frame Equation (22.10) implies the invariant relation g^ v PnPv =r n 2 , which can be written in the form gf* v (P [l -ea [1 )(P v -ea v )=m 2 by using equations (22.5) t h r o u g h (22.9) and the relation PH=g^vPv for CLA frames. This equation for the canonical momentum P^ of a classical charged particle suggests that the generalized Klein-Gordon equation for a quantum charged particle in a CLA frame should have the form

g ' x v ( i D ( 1 - e a ^ ) ( i D v - e a v ) - m 2
i.e.,

g^v = ( w - 2 , - 1 , - 1 , - 1 ) ,

W" 2 (iD 0 - e a 0 ) 2 - (iD! - e a i ) 2 - (iD 2 - e a 2 ) 2

- ( i D 3 - e a 3 ) 2 - m 2
(J = l)

Note that the canonical momentum P^/i must be replaced by

(22.15)

covariant

derivatives D^ as defined in the Riemannian geometry with a metric tensor g^y

334

Principle of Relativity in the Physics of Accelerated Frames

given in (22.3).

Similarly, the generalized Dirac equation for a CLA frame F

should have the form

r^xHP^-ea^-m^O,

or

(22.16)

[W- 1 Y 0 (Po-eao) + Y 1 (Pi-ea 1 ) + Y 2 (P 2 -ea 2 ) + Y 3 (P3-ea 3 )-m] 1 ) ) = 0,

where

P ^ (P 0 ,-P), T|> - i>(w,x,y,z);

{y%,y]g}=rfb,

I*(x)-(wW,?2^3),

n a b = (1,-1,-1,-1),

(22.17)

a,b = 0,1,2,3.

The explicit form of the operator P^ for the Dirac spinor turns out to be slightly different from that for a scalar function.

(See equations (22.19) below.)

Evidently, the generalized Dirac equation (22.16) reduces to the usual Dirac equation in the limit of zero acceleration, <x0 -» 0 or W -» 1. If one wishes, one can relate r ^ x ) in (22.16) to the constant Dirac matrices y a . a = 0,1,2,3, using the relationship r fl (x)= e^(x)Ya, where e|J(x) is a tetrad (i.e., a unit tangent vector) and satisfies the relations

2

eg(x)e£(x) = ^

and

a-0

^ e a | i ( x ) e a v ( x ) - g^.

(22.18)

a=0

Note that the subscript a is of no significance to the covariance. When e =0, we have the "free" Dirac equation in a CLA frame involving a "gauge covariant differentiation":

(ii^-m^-Q,

V^Lo+ifekWhSYDL

ai, 92, a 3 ),

Dynamics of Classical and Quantum Particles

( 22 -19)

335

which will be discussed in section 23 c. In the absence of the electromagnetic potential a^(x), or when the potentials do not involve time explicitly, one can use the separation of variables in (22.19) to find the w-dependent part of \p : ^-F^w^r),

(22.20)

C -i

w

?

)

Fw = F w 0 exp { — J" Y (w')Mdw' J Yo 0

F

expj^Lln (1 + P ) ( 1 " P o ) l ( l - P ) a + Po) 2a0Yo

(22 21)

where M is a constant, p=ct0w+p0, and we have combined the space-dependent terms involving the same Dirac matrices together (e.g., YkW-1dk'W and Ykdk> k=l,2,3).

Note that (22.21) reduces to the usual form F w =F TO exp(-iMw) in the

limit of zero acceleration, a 0 -> 0. 22c.

Stability of

atomic

levels

against

constant

accelerations

A physical system is described by a Hamiltonian (e.g., P 0 in (22.16)) and the Hamiltonian in our formalism for accelerated frames has the dimension of mass. Thus, let us now consider the "mass levels" of a hydrogen atom at rest in a CLA frame F.

First, we must determine the generalized Coulomb potential

produced by a charged particle at rest in a CLA frame. Such a Coulomb potential can be obtained by solving the generalized Maxwell's equation g^D^fya = j a , (or g^fD^EVaa-D^Daav] = j a ) given by (22.14) for a CLA frame with the 4-potential a^ = (a 0 ,0,0,0) and the current density j ^ = (e 6(r),0,0,0) in a CLA frame. Based on the covariant differentiations in (22.12) and (22.14), and the metric tensor g^ v in (19.18) and that in (19.8), one can verify that D^DaAv = DaD^Av for an arbitrary vector A v . (This is related to the fact that the RiemannChristoffel curvature tensor R v 0 j [ i a v a n i s h e s , (D^D a -D a Dn)A v = - R v a , n a A a = 0, for the spaces of CLA frames with both the metric tensors given in (19.18) and

336

Principle of Relativity in the Physics ofAccelerated Frames

in (19.8).) As usual, we can choose a gauge condition E^ay = gv^E^av = 0 to simplify Maxwell's equations.

Thus, we have

g^vD[iDva|X =j a .

When

a(i(w,r)=(a0,0,0,0), this equation leads to the following generalized Coulomb equations for CLA frames: g ^ D v a o = g^Vvao - ^ - g ° ° a 0 a 0 + — ^ — a ^ o - j 0 1-P Y5 +a o x for

g^v=(Y4(Yo-2+aoX)2i_1)_1|_1)) (22.22)

a g^vD^Dva0 = g>"Vv 0+ r

a

\

l + Yo«

for

x

^0 = j 0

gM.v=((i+Voa*x)2i_1>_1)_1)>

In general, these equations give complicated potentials produced by a simple point charge. Let us consider the static case, a0 = a0(r) and j 0 =e 5(r), with the first order approximation, ao/^o^+a^)"^.

Under these conditions, both

equations in (22.22) lead to the differential equation and the solution for the generalized Coulomb potential in CLA frame: (v2-a0—)a0 =-e8(r), 3X

(22.23) a

e -(l i4KT{

ou = —

+

•

2fi*

i.

2J

As expected, they become the usual equation and Coulomb potential in the limit of zero acceleration, ci0 -» 0.

Dynamics of Classical and Quantum Particles

337

The generalized Dirac equation for the electron in an

accelerated

hydrogen atom (i.e., at rest in a CLA frame) is given by (22.19) with rn-given by (22.17) with the usual replacement, iv^ — (iv^- ea^) [for e<0, J=l]. We have

1

WJw

2W

flw

W 4irr {

2 )

+aDxi —+ a Dv i — + a Dz i y ax by dz

pm W = 0, '

(22.24)

where a D = (aDX, aDy, aDZ) = (PpY1, PDY2> PDY3) and (3D = y° are the Dirac matrices. If one uses the W given by (22.2), i.e., W = Y2(YO2 + a o x ) associated with the Wu transformation, the effective

potential e a g / W = -(e 2 /4jtr)(l + a 0 x / 2 ) / W

becomes time-dependent due to acceleration of the whole atom because y involves w.

In this case, the solution is more complicated due to the w-

dependent W. On the other hand, suppose one uses the time-independent W given by (19.8), i.e., W = W x = (l+v0a*x) which is related to the generalized MWL transformation.

The effective

Coulomb potential

eao/Wx

is t h e n

also

time-independent and one has a stable atom, just as in an inertial frame. In most cases, the generalized Dirac equation in (22.24) is complicated and does not have the usual spherical symmetry because of the presence of the metric tensor g^v and the generalized Coulomb potential given by (22.23). However, the violation of spherical symmetry turns out to be extremely small and the Dirac equation for the hydrogen atom can be well approximated by the usual form in an inertial frame. _8

the atomic domain, r~10 cm.

The reason is that we are interested only in We have at present linear accelerators in an

inertial frame Fj (laboratory) with a maximum voltage gradient of about 70 MeV per meter. We estimate the acceleration of a hydrogen atom to be a*~0.05/m for

338

po~0.1.

Principle of Relativity in the Physics of Accelerated Frames

(22.25)

Thus, the extra x-dependent part in (l+y 0 a*x) and the generalized Coulomb potential given by (22.23) is much smaller than 1, a*x~10-n,

(22.26)

where x is roughly the size of the atom, x~10~8cm. Therefore, the violation of spherical symmetry in the generalized Dirac equation is negligible for all practical purposes. As a result, the generalized Dirac equation in (22.24) can be well approximated by the usual Dirac equation, so that the mass levels of an accelerated hydrogen atom are given by 1

M(n);

) + -,

2

4

-fT^

n-h- •

,hZ

(22.27)

2

~2

where a e = e 2 /(4JT) (~ 1/137) in natural units.

Thus, the atomic mass level

structure is relatively insensitive to linear accelerations.

When an electron

jumps from a state nj to another state n 2 , it will emit or absorb a 'mass quantum' k 0 (with J=l): M[(n 2 ) - M i U i ) = k r o ,

inFi,

(22.28)

M(n2)-M(n1)=k0,

in F.

(22.29)

If two photons with "moving masses," kio and ko, are emitted from two h y d r o g e n atoms at rest in Fi and F respectively and measured immediately, then the results (22.27) through (22.29) imply ko(rest) = k I0 (rest),

(22.30)

where the "(rest)" associated with ko and kio refers to the state of motion of the source of photons. The result in (22.30) is isotropic when the photon is emitted

Dynamics of Classical and Quantum Particles

339

and immediately measured in F. On the other hand, if the measurement is delayed, then the acceleration of the F frame will cause a Doppler-type shift. 22d. Electromagnetic fields produced by a charge with a constant-linear-acceleration In 1909, Max Born first discussed the motion of a charge with a constant linear acceleration based on special relativity.4 In the case where the motion of the charge is along the xi-axis of an inertial frame Fi(wi,xi,yi,zi), Born obtained the hyperbolic motion (18.4), i.e., r-2

r-2

Xi=T/(c2/g)2+(cti)2-^-,

V l

--,

8

C t;

(22.31)

V(c 2 /g) 2 +(ct,) 2

This motion can be viewed as the motion of a particle under a constant external force, Fiext=const., along the x-axis, - f Pi - F,ext,

pi = ,

dt

mVl

,

v, =(v,,0,0),

(22.32)

Jl-v 2 /c 2

i

in special relativity. Born also obtained the electromagnetic fields associated with this type of motion of a charged particle. Let us compare the motion described by (22.31) with the motion implied by the Wu transformation (19.14) for a CLA frame F(w,x,y,z). Suppose a particle is at rest in the CLA frame F at X=XQ and y=z=0. Its position (xi,0,0) and velocity dxi/dwi can be expressed in terms of the time wi in the inertial laboratory frame Fi(wi,xi,yi,zi),

xi =,1x0+—L— "oYoJ

340

+wI+-^2{

«oYoJ

^—, a

°Yo

Principle of Relativity in the Physics of Accelerated Frames

(22.33)

dxi dwi

:

W] +"oYo

a 0 W + Po

x = XQ ,

(22.34)

Wj + -

x0 + o I

for

*oYo

2

«oYo

where we have used the Wu transformations (19.14) and (19.15). We observe that there are non-trivial dependencies on the initial velocity p 0 and position x 0 in the CLA motion in (22.33) and (22.34). The initial position xi(0) at wi =0 can be obtained from (22.33),

\2 W] = 0 ,

xi(0) = «oYo

= xn

(22.35)

"oYo

"oYo

for

p o =0,

Yo = l-

From (22.31), (22.33) and (22.34), we see that (22.31) corresponds to the special case of (22.33) and (22.34) when both the initial position and the initial velocity of the F frame are zero, x 0 = 0 and pQ = 0.

However, an important

difference is that the acceleration corresponding to the hyperbolic motion described by the equations in (22.31) is d 2 xi/dt 2 = (c 6 /g 2 )/[(c 2 /g) 2 +(cti) 2 ] 3 / 2 , so that in the limit of zero acceleration g->0 one has the result xi -»0. It does not reduce to a constant motion with non-zero velocity.

Note that the constant

acceleration of a particle in special relativity is defined in its instantaneous rest frame V! = 0, i.e., ti = 0 from (22.31) and hence, d ^ / d t i 2 = g = constant.

On the

contrary, the limit of zero acceleration of the motion implied by the Wu transformation has a well-defined constant motion: xj = (XQ+YOPOWI)/YO as a 0 -=• 0. In other words, in the limit of zero acceleration, we obtain exactly the Lorentz transformation with the constant velocity p0- The reason for this difference is that the CLA motion in (22.33) and (22.34) satisfies the limiting 4-dimensional symmetry, while the hyperbolic motion (22.31) in general does not. In the CLA motion, the acceleration of this particle can be derived from (22.34)

Dynamics of Classical and Quantum Particles

341

\2 d 2 X!

"oYo

3/2

dwf

x = x0-

(22.36)

Po

WJ + a

"OY£

, for

oYoj

Thus, the Wu transformation implies that the acceleration of this particle depends on its position x0 and the initial velocity p0 of the CLA frame. The relativistic equation of motion (22.32) cannot give this detailed information. As a result, if one uses (22.31) or (22.32) to derive transformations for reference frames with constant linear acceleration, one obtains the 4-dimensional group of conformal transformations in spacetime,5 which turns out not to be physically meaningful and does not satisfy the limiting 4-dimensional symmetry. Now, however, we have the much more satisfactory Wu transformation which does satisfy the limiting 4-dimensional symmetry and also provides more detailed information concerning accelerations. We now apply it to the investigation of the electromagnetic fields produced by an accelerated charge, classical radiation and energy conservation. First, let us consider the electromagnetic fields produced by a charge e , whose motion is described by (22.33) and (22.34). According to Maxwell's equations, the retarded 4-potentials ai^(f) at the field point f are the covariant Lienard-Wiechert potentials 6 as observed in an inertial frame Fi eu^(s) ai,x(f) •

f s (w/, x,f, y,f, z,f ), Un(s) = V uV < s )>

342

(22.37)

4It(Xf-xf)xu?l(s) s s (w,s, xf, y,s, z,s),

'HV

-(1,-1,-1,-1),

Principle of Relativity in the Physics of Accelerated Frames

where s denotes the source point and u v (s) the 4-velocity of the charged particle (the source) , v . dw^ dx* . u v ( s ) - (v— J - , - j J - , 0 , 0 ) , ds ds "

Wj +-

dwj

"oYo/l

«oYoj

~dT

(22.38)

a0Yo

, s dxi ds

. s,. s dxi/dwj

wf+—!-5I

"oYo

Vl^ «oYo

where we have used (22.33) and (22.34).

Note that the charged particle is

assumed to be at rest at the position (xo,0,0) in the CLA frame F(w,x,y,z), i.e., r s = (Xjs,0,0) in the inertial frame Fi. It is important to note that the field time w,f is related to the source (or emission) time wf by the relation w , f - w f = lr, f - r : s l > 0.

(22.39)

This is the causality condition that specifies the amount by which

the

Lienard-Wiechert potential (22.37) must be retarded in order to be consistent with experiments and observations. To find the potential a ^ f ) at the field point f =(Wjf, Xjf, y T f , Zjf), we must express the source coordinate (Wjs, Xjs, yj s , Zjs) in terms of the field point f. This can be done because the square of the causality condition (22.39) enables us to solve for wi in terms of (w/ , Xj , Vj1, Zj')

Dynamics of Classical and Quantum Particles

343

(wJ-w^-Crf-rf) 2

xf-,

wf+^2a 0 Yo

"oYo

(22.40)

+P

a 0 Yo

P2 = (x,f- y : s ) 2 + (z, f -z, s ) 2 = (y,f ) 2 + (z,f ) 2 . For calculations, it is convenient to set

A^Ja."

Wf = W, +

,

W s = W js +.

«oYo

X f = x if +

Po aoYo

1

,

ki = x 0 +

«oYo

1 -. aov 2 ,

It is then straightforward to obtain

,f+_Po_) A*-fxf+-J-)B* a

«? J ._Po_

oYo)

\

«oYoI

a

I

a

(22.41)

*oYo

oYo ^

oYo;

A* = I1 xf + —i—] +p2 + k2- f w f + -2e_) , a0Y0j

\

2

f

a

oYo

a0y0J

_f x f + _i_f- p 2 +k 2 'I

a

+4k2p2 .

oYo

Similarly, we can also express the coordinate Xj of the charged particle in terms of the field point f,

344

Principle of Relativity in the Physics of Accelerated Frames

X ? + ^ = Jk2+[w?+J2_ a0Yo

2

x'L,fI .

a

1 |A*- w oYo

B* a 0 Vo

(22.42)

«oYo "oYoj

I

a

oYo

where we have used (22.33), (22.34) and (22.41). It follows from (22.38), (22.41) and (22.42) that

(x[-xf)^U(i(s) = -

B*

(22.43)

aoYo Thus, the retarded potentials at ^(f) in (22.37) produced by a charged particle at rest at a point (x o ,0,0) in the CLA frame F are given by

B* ,xf+_A_jA*-lwif._Po__ +

a ro (f) - _e_ J, An B*

« 0 Yo)

\

«oYo)

w

e I

an(f) = - aii(f)

4re B*

"oYo

\

(22.44)

«oYo)

f_Po_A*_xf+^_|B* a0YoJ \ "oYo;

(22.45)

1 f "oYoy

«oYo J

ai2(f) - a B (f) = 0 .

(22.46)

The electromagnetic field E(f) can be calculated from equations (22.44) t h r o u g h (22.46). Wehaye

Dynamics of Classical and Quantum Particles

345

Xo+-i-)2-f4+^f+fwf+^)2+p2

En(f) = *(B*)

En(f) =

a

J

aoVo

2e

f

ji(B*) J

E3(f) = 2e n(B*f

f x

y\

«oYo

Z

f I

X

f I+

"OYJS

oYoj

I

aoYo/l

I

"oYo/

1 i +"oYo —

(22.47)

1 — a 0 Yo

Similarly, the magnetic induction B,(f) is found to be Bn(f)-0,

Bi 2 (f) =

BB(f) =

z\U\+-^-) a

-e x(B*f

«oYo

n(B*f

"oYo

oYoJ'

I

(22.48)

"oYo

In these calculations, we have used

Ei(f)

aai aaI0 aw i ari

B,(f) = [vxa,] f

(22.49)

where [ ]f denotes that the quantities within the brackets [ ] refer to the field point f. The function B* is given by (22.41) and the coordinates of the charged particle in F[ are (xpypzj1) = (Xj,0,0). One can verify that Born's results

346

Principle of Relativity in the Physics of Accelerated Frames

correspond to the results (22.44)-(22.48) in the special case XQ = 0 and p 0 = 0. Although these expressions involve l / a 0 , they do not diverge in the limit of zero acceleration a 0 - » 0 .

As a matter of fact, in the limit a 0 ^> 0 , the retarded

potentials in (22.44)-(22.46) reduce to the usual ones found in textbooks.

6

standard

Moreover, the electromagnetic fields in equations (22.47)-(22.48)

vanish at the "black wall" x 0 = -l/a 0 Y 0 2 , the singularity wall of the CLA frame. The causality condition (22.39) can be expressed in terms of the field point f=(w[,x[,y[,z[),

x U - U _fwf+_Bo_ «oYo J

a

[

oYo

xU-L.)

x f x f + _l_Wf w f + -Pa«oYo)

XfB* Wf x^2 - W2j

[

«oYo

W

f+J^

,p2+k2

«oYo

+p 2 + k 2

«oYo;

«oYo

• 0.

(22.50)

«oYo

When a 0 -»0, it reduces to

f s 2 w| -W»=Yo

xf-^|Po"PoW[

+

Vol U f - ^ - P 0 w f

+

p 2 >0.

(22.51)

Yo

Thus, the restriction on (w[,x[,y[,z[) to satisfy the causality condition is not simple even in the limit a o -»0. According to classical electrodynamics, an accelerated charge emits electromagnetic radiation.

The radiation rate for a charge with

arbitrary

motion can be calculated using the Lienard-Wiechert potentials (22.37):

Dynamics of Classical and Quantum Particles

347

„ rad

"3

2^ 2 du^(s)du^(s) ~di ds~~'

(22.52)

This holds generally for any source point s with the retarded velocity 4-vector u^-(s) = (dwf/ds, dxf/ds, 0, 0) of any source point s.

For the CLA motion of a

charge, the acceleration 4-vector duM-(s)/ds can be obtained from (22.38). We have

du°(s) ds

"oVo

(22.53)

aoYo

du*(s) ds

«oYo

aQYo

(22.54)

«oYo

du 2 (s) ds

du 3 (s) ds

0.

These lead to the relation chys) ds

d u t x( s )

(22.55)

ds "oYo

Thus, the radiation rate for a charge with a CLA motion is a constant

Rratl =

2e2 i2

«oYo

348

Principle of Relativity in the Physics of Accelerated Frames

(22.56)

The radiation rate Rrad has the dimension of 1/length 2 because e is dimensionless in natural units, as given by (22.12), and x 0 has the dimension of length. We can thus estimate the radiation rate for an electron at a high energy laboratory such as the Stanford Linear Accelerator with wi=ct and a potential gradient of 60 MeV per meter. Setting Xo=0, po=0 and using e in (22.12), the result (22.11) with (dpio/dxi)=60MeV7c2 per meter, and the electron mass m=0.5MeV/c2-10-30Kg, we obtain R r a d » - e 2 a £ - - 10"40 Kg/m - - 0.3 xlO3 / m 2 ,

(22.57)

which corresponds to about 10 -23 watts in conventional units. This is extremely small compared to loss of energy by an electron moving in a circular path. For example, at the 10 GeV Cornell electron synchrotron with an orbital radius of 100 meters, the loss of electron energy per orbit is about 9 MeV,6 which corresponds to about 10 -2 MeV/m or 10~6 Watts. This is a significant loss of the electron energy. Thus, r-f power must be supplied to maintain electrons at a constant energy as they circulate. For an electron with CLA motion, the loss of energy is completely negligible at the presently available accelerations. 22e. Covariant radiative reaction force in special relativity and common relativity A complete satisfactory treatment of the radiative reaction forces and their effects exists neither in classical electrodynamics nor in quantum electrodynamics. This is a profound difficulty which is intimately related to the conceptual framework of spacetime and elementary particles. Whenever a charged particle is accelerated by an external force, it emits radiation which carries away linear momentum and angular momentum. Thus, the emitted radiation influences the particle's subsequent motion. However, the situation in special relativity is not satisfactory because the covariant radiative reaction force vanishes for a charged particle which moves with a constant linear acceleration and radiates. In contrast, common relativity does not have this problem and is more satisfactory.

Dynamics of Classical and Quantum Particles

349

In the following discussions in this section, all coordinates and momenta are those of the charged particle (i.e., the source of electromagnetic emission) in a general inertial frame. The superscript s for the source and the subscript i for quantities measured in an inertial frame will be suppressed in this section. Only the constant (x o ,0,0) refers to the position of the particle in a CLA frame. First, let us consider the covariant radiative reaction force in special relativity. The usual form of the equations of motion such as (12.4) and (22.12) is not completely satisfactory because they do not take the radiative reaction force into account.

Once the radiative reaction force appears, the system becomes

non-holonomic because such a force cannot be derived from a potential in the Lagrangian formalism. The Abraham-Lorentz model of an electron with a finite size has been studied in detail and leads to the Abraham radiative reaction 4-vector. 7

Dirac's formalism of Lorentz covariant classical electrodynamics 8

(c=l) also gives the same result

dp^ - F11 + F^ ds - ^ x t + W -

(22.58)

F&t = e F ^ u ,

(22.59)

'dV rad

3m l^ds2

p*1 d p a d p a ' , 2 ds ds

(22.60)

Let us consider the CLA motion described by (22.33) and (22.34), which includes hyperbolic motion as a special case. Since p^ = mu^, differentiating eq. (22.38) leads to

1 dV

a

m ds 2

oY 0/ J

a

Po oVo \3

a

350

Po

w+ "oYo

oYo/

, 0, 0

«oYo

Principle of Relativity in the Physics of Accelerated Frames

(22.61)

\2

m

a

oYo/

This second order derivative of momentum may be called the "jerk" 4-vector, since the time derivative of the acceleration is known as jerk in mechanics. Although this quantity "jerk" never appears in the fundamental laws of nature (i.e., elementary particle physics or quantum field theories), it is relevant to robotics design and in tracking systems for fast moving objects. From (22.54), (22.60), (22.61) and pl1=mu^ we have the following result based on special relativity (c=l):

r

rad

2e z 3m

(-P11)

aoY^

(22.62)

"oYo

This implies that for CLA motions, the radiative reaction force vanishes, so that equation (22.58) turns out to be the same as the conventional equation dpM-/ds= e F^vuv, even though the charge emits electromagnetic radiation. This is indeed a very strange result. In the literature, Pauli and von Laue said that charges with CLA motion (i.e., the hyperbolic motion) do not radiate.^ If they were right, the result (22.62) would be satisfactory.

However, their statement is incorrect. 5

Other physicists such as Langevin, Poincare and Heitler have concluded that there is indeed radiation.4 The result (22.60) for the radiative reaction force of Abraham and Dirac appears to be unique because any force must satisfy the relation p^Fn=0 which follows logically from the basic 4-dimensional law ds2 = dx^dx^, x^ = (ct,x,y,z), and the definitions p^ = m # and c = 1 in special relativity. Furthermore, the 4-dimensional symmetry of the constant linear acceleration requires that the jerk 4-vector be proportional to the velocity 4-vector, as

Dynamics of Classical and Quantum Particles

351

shown in (22.61).

So the situation appears to be quite hopeless because

4 - d i m e n s i o n a l symmetry of special

relativity

seems to be

exceedingly

restrictive. Fortunately, common relativity offers a way a r o u n d this

difficulty. 2

Within the framework of common relativity, one has the basic law ds = dx^dxt1 with xl* = (bt c ,x,y,z), but the common time t c is an invariant quantity, just like ds. Consequently, one also has p^p^ = m 2 , where pH- = mdx^/ds. By differentiating p(ipvt=m2 with respect to the common time t c twice, one obtains ( d p ^ / d t j p ^ = 0 and (d2p»Vdt2 ) p (x +(dp^/dt c )(dpn/dt c ) = 0.

These relations imply that the

covariant radiative reaction force has the form

dV.PLdp.dp^ dt 2

(2263)

m 2 dt c dt c

Note that in the absence of the radiative reaction force Frtfd in (22.58), the equation of motion, mduf*/ds= e F , i v dx v /ds, can be written in terms of i n v a r i a n t common time t c , dpiVdt c = e F^ v dx v /dt c .

Thus, in the framework of common

relativity, we can postulate the basic equation of motion for a charged particle emitting radiation to be

"P

dt c

F^

-TV

eXt

4. P 1

rad

'

- o f 1" " X v

tfiXt

"6t

Fn rad

_ 2 e dtc d V , P* dp a dp a l 3m ds dt 2 ' m 2 dt c dt c

dt c '

(22.64)

(22.65)

(22.66)

The invariant factor dt c /ds in (22.66) is necessary for the dimensions to be correct.

352

Principle of Relativity in the Physics of Accelerated Frames

Let us now consider the conservation of "energy" p° in the general equation of motion (22.64). The zeroth components Fext and Frad in (22.64) are respectively the rate of work done by the external force (i.e., dWext/dt c ) and by the radiative reaction force, as measured using common time t c . Since d/dt c = (ds/dt c )(d/ds) = (C 2 -v 2 ) 1 / 2 (d/ds), where C = d(bt c )/dt c , we have

dV dt 2

|

p» d p a d p a m 2 dt c dt c

=

V^ 2 ~U-Vc 2 -v 2 fe + ( c 2 - v Ids J ds \

a dv a dp" = v ds ds

d 2 p" ds'

p<* d p a d p a i 2 ds ds

v-f-(Cv),

(22.67)

where we have used (22.61). The result in (22.38) gives

v « = ^ = f - ,—,0,0

«oYo;

0, 0

(22.68)

Po «oYo/

«oYo

From equations (22.66) through (22.68), we obtain a non-zero radiative reaction force in common relativity

Fo rad

_ 2e z dt c v „ dv a dp*1 3m ds d s ds

Dynamics of Classical and Quantum Particles

353

.211

«oYo

(22.69)

3m x0 +

Po

x0 + a

"oVo

oVo;

«oYo

where we have used dC/dt c = 0 by choosing the inertial frame to be the one i n which the speed of light is constant. This can be done because v a ( d v a / d s ) is an invariant in common relativity. Note that F^d never positive, as expected. In general, the zeroth component of (22.64) can be written as the law of conservation of "energy" (with the dimension of mass) for an arbitrary motion, d p 0 _ dW ext dt r dt r

dW,rad dtr

dW r a d

(22.70)

dtr

Cdtr

2e 3

duu dt c

.21 "m

-CR rad

(22.71)

where R r a d is a scalar quantity given by (22.52). Its physical meaning is that the rate of change in the kinetic energy of a particle is equal to the rate of work done by the external force minus the rate of work done by the radiative reaction force.

The work done by the radiative reaction force is always negative and

involves two parts: (i)

One part is related to the rate of change in (2e 2 /3)(du°/dt c ) which

may be called the "acceleration charge energy." energy" has discussed by Schott. 4

The idea of "acceleration

This "acceleration charge energy"

is

independent of the sign of the charge and depends only on the rate of change of u° = Y, just like the energy p° of a particle. (ii) The second part is related to the radiation rate R r a d and is n e v e r positive.

354

Principle of Relativity in the Physics of Accelerated Frames

References 1.

J. P. Hsu and Leonardo Hsu, Phys. Lett. A. 196, 1 (1994).

2.

J o n g - P i n g Hsu and Leonardo Hsu, in Symposium

JingShin

Theoretical

Physics

in Honor of Professor Ta-You Wu (Ed. J. P. Hsu and L. Hsu, World

Scientific, Singapore. New Jersey, 1998) pp. 393-412; Chinese J. Phys. 35, 407 (1997). 3.

C. Moller, The Theory of Relativity (Oxford, London, 1969) pp. 264-287.

4.

M. Born, Ann. Physik 30, 1 (1909); A. Sommerfeld, Ann. d. Phys. 3 3 , 670 (1910). G. A. Schott, Phil. Mag. 29, 49 (1915). The electromagnetic field produced by a single accelerated charge was discussed by Langevin in 1905, Journal de physique 4, 165 (1905).

Following Langevin, Poincare first

discussed the Lorentz covariant retarded potential in his Rendiconti paper in 1905. (For its English translation, see H. M. Schwartz, Am. J. Phys. 40, 862 (1972).)

He found that there were terms in the electromagnetic field

involving the acceleration of the charge and this "acceleration wave" (as it was termed by Langevin) became dominant at large distances. Also, the E and B fields of the "acceleration wave" satisfy IEI=IBI, E-B=0 and E-n=B-n=0. For these terms involving acceleration, see W. Heitler, The Quantum

Theory

of Radiation (3rd. ed. Clarendon Press, Oxford, 1954). pp. 20-25. 5.

T. Fulton and F. Rohrlich, Ann. Phys. 9, 499 (1960).

6.

J. D. Jackson, Classical Electrodynamics

(2nd ed. John Wiley & Sons, New

York, 1975) pp. 654-662. 7.

J. D.Jackson, ref. 6, pp. 786-791.

8.

P. A. M. Dirac, Proc. Roy. Soc. A 167, 148 (1938); See also Jackson, ref. 6,

9.

W. Pauli, Theory of Relativity

p.808. (Pergamon Press, New York, 1958) p.93. "For

the electromagnetic fields produced in the frame in which the field point and the center of the hyperbola are simultaneous," Pauli obtained the result that "there is no formation of a wave zone nor any corresponding radiation." But he also said parenthetically that radiation does occur w h e n two uniform,

rectilinear

motions are

connected by

a "portion" of

hyperbolic motion.

Dynamics of Classical and Quantum Particles

355

23. Quantizations of Scalar, Spinor and Electromagnetic Fields in Constant-Linear-Acceleration Frames 23a.

Scalar

fields

in

constant-linear-acceleration

frames

In constant-linear-acceleration (CLA) frames, physical equations and calculations are generally more mathematically complicated than those in inertial frames. The reason for this is that the differential time-axis dw in CLA frames is altered by the Wu factor W(w,x)=Y2(Yo~2+aox) > 0. a s shown in (22.2), so that the physical space is bounded by a "black wall" at x=-l/a 0 Y 0 2 - This "black wall" is a wall-singularity where the speed of any physical signal or particle becomes infinite.

Nevertheless, the CLA frame

with a constant-linear-

acceleration is the simplest non-inertial frame, and offers the best opportunity to explore physical properties of quantum fields beyond inertial frames or the framework of relativity theory. The investigation of quantum field theory in CLA frames can shed light on more complicated non-inertial frames and perhaps on quantum gravity. Furthermore, when considering the S-matrix and the Feynman rules

in

quantum field theory, we must use a plane wave (for a free particle) which is very simple in inertial frames, but complicated in CLA frames.

As a result,

many calculations in CLA frames are most easily carried out by a change of variables. Such a change of variables effectively transforms calculations from a CLA frame to an inertial frame.

Of course, there are non-trivial problems

which must be solved in CLA frames, such as finding the Coulomb potential generated by a charged particle at rest in a CLA frame and solving for e n e r g y states in such a potential, as discussed in chapter 22. Because we have specific coordinate transformations between inertial frames and CLA frames, we can explore their implications for possible new physics of particles and fields in non-inertial frames. The equation of motion of a classical particle in CLA frames has been discussed in chapter 22. Even the simplest motion, e.g., constant velocity in an inertial frame, will have more

356

complicated properties from the viewpoint of a CLA frame, as shown in the momentum transformations in (22.10). A similar complication occurs in field theory:

The Klein-Gordon

equation for a "free" (i.e., non-interacting) particle in a CLA frame can be obtained from (22.15) by setting e=0: [g^ v D (l D v+ m 2 ]
(23.1)

where x denotes the position vector and D^ is the partial covariant derivative as defined in (22.12) with the fundamental

metric tensor given

by

(19.8)

(associated with the generalized MWL transformations), gjv =(W2,-1,-1,-I) = ([l + Y0 a*x] 2 ,-1,-1,-1),

(23.2)

or given by (19.18) (associated with the Wu transformations)

g^v =(W 2 ,-1,-1,-1) = ( Y 4 [l + a 0 x] 2 ,-1,-1,-D.

(23.3)

In the following discussion, the form of physical equations (or laws) involving W hold both for metric tensors in (23.2) and in (23.3). In other words, the function W stands for either [1 + y0a*x] >0 or Y2[YO"2 + a 0 x] > 0. Using the covariant expression for d'Alembert's operator gi^D^Dy we find that

g ^ D ^ o - -jLa^VIg" g^flvo) -g

{w" 2 fl 2 -v 2 -W- 3 (a 0 W)ao-W- 1 (fl 1 W)ai}*,

(23.4)

V^g" = 1 / - d e t g o p = W > 0 ,

Quantizations of Scalar in Constant-Linear-Acceleration Frames

357

where

dVb=(d0,di,d2,d3)

X = (x 1 , X2, X 3 ).

= (S/dW, d/dX),

(23.5)

The partial differential operators (23.5) are related to those in an inertial frame a

in =(^io»aii'ai2»ai3) = (3/aw!, a/axj, a/ax 2 , a/ax 3 ) by the relations,

a I 0 = Y (W- 1 a 0 -pa 1 ),

an = Y (a 1 -pW- 1 a 0 ),

sI2 = d2,

a I 3 =a 3 ,

(23.6)

which can be derived from the transformations in (19.14) (or (19.4) with W i n (23.6) replaced by W x ). The covariant derivatives a^ transform in the same way as the covariant momentum 4-vector p^ in (22.10), as expected. It follows from (23.6) that a20 - v,2 = W"2a2 - v 2 -W-3(a0W)a0 - W ^ W ) ^ ,

(23.7)

which is consistent with the invariant expression for d'Alembert's operator g ^ E ^ A in (23.4). For CLA frames under the Wu transformations, W is given by (23.3), and the free Klein-Gordon equation (23.1) takes the form,

{ 1 a2_v2_ 2ag£ W2 W2

aa_ (75 2 + a 0 x)

2U

= Q^

(23.8)

The solution of this "free" Klein-Gordon equation in a CLA frame takes the form


XE X f,

P(x) = p I ow I (w,x 1 ) + pI1x}(w,x1) + p I 2X 2 +p I3 x 3

358

Principle of Relativity in the Physics of Accelerated Frames

(23.9)

1

Po

"oYoS

OoYo

1

PlO YP x .

YPO YP x W

+YP1

*-1 «oYo

«oYo

x^-1 a

oYo;

«oYo

+ pI2x2+pI3x3

+ PI1 Y a

oYo/

-YPPl YP x!

aQYo

1

Po

+ a

a

oYoJ

YPPO W

X * -

oYo

+ pI2x2+pI3x3)

1

«oYo;

"oYo

where W = WCw.x1) and we have used (22.10). The phase P(x) reduces to the usual invariant phase pow + p i x 1 + P2X2 + P3X3 in the limit of zero acceleration a^O. Note that pio and p n are constant momenta measured in F] and that t h e y are related to the non-constant momenta p^ in the CLA frame F by the transformation given in (22.10).

The "free" wave function (23.9) with t h e

complicated phase P(x) in a CLA frame is actually a simple plane wave in an inertial frame Fj. If one substitutes the solution (23.9) with the phase P(x) into (23.8), one obtains g^ v PnPv - m2 = ( W W - p 2 - m 2 ) = 0.

(23.10)

This is the invariant relation for the momentum 4-vector, which takes the Lorentz invariant form p ^ v - m ^ O in an inertial frame.

2 3 b . Quantization of scalar fields in CLA frames The canonical quantization of fields mimics the dynamics in quantum mechanics. In fact, it is a generalization of the quantum mechanics of a finite system to an infinite system. In the following discussion, we follow closely the canonical quantization of fields in inertial frames. The invariant action S for a neutral scalar field in CLA frames is assumed to take the form:

S = /Lsd4x,

L s =i A / I g(g" v VM v *-m 2
Quantizations of Scalar in Constant-Linear-Acceleration Frames

(23.11)

359

g = detgap,

J = 1,

where the metric tensors are given in (23.2) or (23.3). The extra function of spacetime V-g

in the scalar Lagrangian density Ls comes from the i n v a r i a n t

volume element V-g d4x=Wd4x. In the conventional field theory formulated i n inertial frames, the Lagrangian density is required to be a functional only of the fields and their first derivatives. In addition, the Lagrangian density must not have an explicit dependence on the spacetime coordinates. 1

However,

when one generalizes field theory from inertial frames to non-inertial frames, these requirements become too stringent and must be relaxed. This is because the fundamental metric tensor g^v(x) is present in the invariant action. As a result, the Lagrangian density will have an explicit spacetime dependence through g^v which stems from the linear acceleration and is not a physical field in our formalism. The geometrical property of spacetime in a CLA frame is completely described by the metric tensor g^v and the Riemann

curvature

tensor can be shown to be zero, implying a flat spacetime without gravity. 2 By varying the scalar field <J> one obtains the free

Klein-Gordon

equation for CLA frames

-j= V ^ - g V-g

g ^ v * ) + m2
(23.12)

which is the same as (23.1). The energy-momentum tensor T ^ for the scalar field is defined by

T

ii=_aLs_^a<j,_svL

=V^g Pw^-^-fg

S^g^oapO-m2*2).

The momentum n(w,x) conjugate to 0>(w,x) is defined by

360

Principle of Relativity in the Physics of Accelerated Frames

(23.13)

jt(w,x) = ^ — V ^

g°°ao*= ^ ao*,

d(d0<s>)

w

x = (xi, x2, x3),

(23.14)

where the presence of the function W in (23.14) is due to the acceleration of the reference frame. The Hamiltonian density Hs and the momentum density P[ of the scalar field are defined by

Hs = ji(w,x)do* - Ls = jyTgig00*2

+ |V
(23.15)

Pi = T° = it(w,x)fli*.

To quantize the neutral scalar field, we postulate the equal-time commutation relation [
(23.16)

0(w,x),(w,x')] = [ji(w,x),jt(w,x')] = 0. As usual, we may express the scalar field operator

(23.17)

5(k(ikM- - m 2 ) = 8(kok 0 /W 2 - k 2 - m 2 )

5(k 0 -Wa) k ) 2k0/W2

|

P - P ( x ) = |Y^Q-YPki)

5(k 0+ W(o k ) 2lk0l/W2 '

YP a

, m2 a)k= Vk 2 +m

1

Po

oYo,

«oYo

Quantizations of Scalar in Constant-Linear-Acceleration Frames

(23.18)

361

+ k2X 2 +k3X 3 ,

+l?kl-Y0^) «oYo

(23.19)

aoYo

where W=W(w,x1) and P(x)=P(xii). The validity of the commutation relations (23.16) for canonical quantization and the Fourier transform (23.17) for the field
density.

This is the reason why it is possible to carry out the canonical quantization of fields in non-inertial frames.

Before quantization, At(k) in (23.17) is the

complex conjugate amplitude of A(k). After the quantization postulate (23.16) is made, one has quantum fields and the amplitude becomes an operator with At(k) the Hermitian conjugate to A(k). Relation (23.18) follows from the formula 3

8(K)(f(x)) = ( - f ) K 6(f(x)) = 2 f——1 v dxy £lf'(xn)l(f'(x)dxj f(x n ) = 0,

n=l,2,3...

o(x-xn), n ''

(23.20)

f(x) = df(x)/dx,

with K=0 and n=2. It is important to note that the quantities (vko/W-yPki) and (yki-vpko/W) in (23.19) are both constants associated with "plane waves," (yko/W - yPki) = constant

and

(yki - ypko/W) = constant,

(23.21)

even though p, y,ko, ki and W are separately non-constants in a CLA frame, as shown in (22.10).

The invariant volume element d4k/-^/g0o = d 4 k/W in the

momentum space of a CLA frame can be obtained from d 4 kj in an inertial frame through the momentum transformation (22.10) with the covariant momentum 4-vector p ^ replaced by k^: d 4 k! = J(ki ^/kx)d 4 k =d 4 k/W,

where J(kix/kx)=l/W is the Jacobian.

362

Principle of Relativity in the Physics of Accelerated Frames

(23.22)

With the help of (23.18), we can write the Fourier decomposition (23.17) for *(w,x) and (23.14) for it(w,x) in the form *(w,x) = f , d k [a(k)e- i p + at(k)e i p ] , 3 V(2it) 2(Dk

jt(w,x) = f

d k it0k

3

[-a(k)e- i p + at(k)ei p ],

a(k) = - ^ L , V2<%

(23.23)

u>k =Vk 2 +m 2 ,

(23.24)

"t) 2cok where k0 in the phase P is equal to Wcok, i.e., k0=wVk2+m2. We can also express the operators a(k) and at(k), which are functions of the spatial momentum vector k, in terms of the scalar field operator O: •3

a(k) = if—.

d x

[eipa0O - (a0eip)O],

(23.25)

3

WV(2ji) 2co k

•3

at(k) = - i f — d x [e-i p a 0 o - (3 0 e- iP )*]. 3 WV(2n) 2u)k

(23.26)

One can verify that the commutation relations [a(k), at(k')] = 6 3 ( k - k ' ) , (23.27)

[a(k), a(k')] = [at(k), at(k')] = 0, are consistent with the equal-time commutators in (23.16). To see this, let us substitute (23.23) and (23.24) into the equal-time commutator (23.16). We have r^i •> , n,= J f, [*(w,x),3i(w,x')] J

d3k J r, d3k' Kitok>

TJ(2n)32akJ

V<2")32cok,

x[a(k)e-ip(x) + at(k)eip(x),-a(k)e-ip'(x')+at(k)eip'(x')]lw=w,

Quantizations of Scalar in Constant-Linear-Acceleration Frames

363

d3k

f d 3 k ' icuk'

f , dk f °~K' lc ° k ' 53(k - k')[e-iP+iP' + eiP-iP']lw_w, J 3 J V(2") 2(o k V( 2jt ) 3 2w k '

=

rjd_k_r J 2(2*)' -w?^3 L

iP+iP' + e iP-iP-]|

J w=w

(23.28)

>k=k '

where the phase functions P and P' are given by (23.19), p = P(X) = P(x,k),

P'= P(x—x', k—k'), x = x^,

k = kn, etc.

Let us calculate the term e _i P +i P' in (23.28). We obtain

e-iP-P'L..., k = k , = e x p [ - i j [ v ^ - y P k 1

YP x l . -

1

a

oiii

+

|Ykl-YP^

xl+-

l

+ k2X

«0Y0

-MlY^-YPk!

YP

\

+ Ykl-YP

«oYo

+k3X 3

«oYo

x*+-i-

«oY0 y "oYo

+ k2x' 2 + k 3 x' 3 ]

x'l+- ! «oYo

«oYo

1 1 1 1 1 = exp[- i ^ j ^ V - X' ) + i ^ k ^ X - x' ) - iySk^X - X' )

+ i L | r < x 1 - x'1) - ik2(x2 - x'2) - ik3(x3 - x' 3 )]

364

Principle of Relativity in the Physics of Accelerated Frames

(23.29)

= exp[-ik-(x-x')].

(23.30)

The other term e i p ~iP' is just the complex conjugate of (23.30). Thus, equation (23.28) is the same as the equal-time commutator (23.16). In general, the Hamiltonian H of the scalar field,

H= j H s d 3 x = / d 3 x - / ^ - [ g 0 V + | V O | 2 + m 2 O 2 ] ,

(23.31)

cannot be expressed in terms of the operators a(k) and at(k) because of the presence of the spacetime-dependent function V-g

= W in the integrand.

Nevertheless, one still can derive physical results from the theory.

The

Hamiltonian H in (23.31) can be expressed in terms of the operators a(k) and at(k) only in the limit of zero acceleration, i.e., W-*l. This suggests that one can define Hw by the relation

HmS f d J

3

x i = f ± d 3 k oj k [at(k)a(k) + a(k)at(k)]. y_g J 2

(23.32)

This Ho, is not exactly the "Hamiltonian" of the scalar field defined in (23.31). It corresponds to p 0 /W and is related to an "instantaneous conservation of energy" in a collision process. Nevertheless, the momentum Pj of the scalar field can be expressed in terms of the operators a(k) and at(k):

Pi = J > i d 3 x = J i d 3 k k ; [ a t ( k ) a ( k ) + a(k)at(k)],

(23.33)

where a(k) and at(k) are respectively the annihilation and the creation operators. We observe that the momentum P; is not constant: [H, Pi] * 0,

(23.34)

in contrast to the corresponding P; in quantum field theory in inertial frames.

Quantizations of Scalar in Constant-Linear-Acceleration Frames

365

The reason is obvious: a constant momentum in an inertial frame can no longer be a constant when it is measured in a CLA frame. The Hamiltonian H in (23.31) contains i~g= W which is an explicit function of space and time. Only in the limit of zero acceleration does (23.34) vanish as it must. One can verify that the operators P(i=(P0,Pi)=(H,Pi) and M ^ generate translations and rotations in CLA frames: IfPn.OJ-an*,

(23.35)

ifM^,*] = (x^v - xvdn),

(23.36)

where Mf*v is given by M(™= f d3x-fg\T0vx»-T°Wl

(23.37)

The Heisenberg equations of motion are given by

i * = i(H - HO) = i [
aw

l

— =-r[jt,H],

aw

(23.38)

! JI=JI(W,X).

(23.39)

'

As expected to insure consistency of quantization, equations (23.14), (23.16) and (23.39) reproduce the field equation (23.12).

23c.

Quantization of spinor fields in CLA frames The previous discussion of canonical quantization of scalar fields in

non-inertial frames with constant linear acceleration can be applied to other fields.

In this section, we will give a brief derivation of the main results for

spinor fields. The invariant action S^ for a free electron field i|> in CLA frames is

366

Principle of Relativity in the Physics ofAccelerated Frames

assumed to be given by

S^/L^x, Ly-^^ymd^-^Tgid^rKy-'srgnm,

rn^w-V.Y 1 ,? 2 ^ 3 ),

{nsni = 2gnv(X), V^i" = w>0, where Y^ = {i0^1,^2^*)

af

(23.40)

e the constant Dirac matrices. The presence of the

metric tensor gi"(x) is a new feature in the formalism of quantum field theory in CLA frames. The fermion Lagrangian must be symmetrized because r^ is now a function of spacetime rather than a constant. Also, the non-universal speed of light does not appear in the Lagrangian density L^. Thus, with the help of integration by parts, the Lagrangian in (23.40) can be written in the usual form in terms of a "gauge covariant derivative" v^,

Li|j =V^g ^rwa^ti) + 2^[i(a |i >n'ri i )]Ti> - \Tgmw>

=Woj)rWV^ - W m « ,

(23.41)

v ^ = ( a 0 + 2
The presence of the gauge covariant derivative can also be seen explicitly if one introduces vierbeins (or tetrads) to express the matrices TV-. Four mutually orthogonal unit vectors, denoted by eav, a=0,l,2,3 form an orthonormal tetrad, where v is the covariant tensor index and the Latin suffix "a" is a label distinguishing the particular vector.

The contravariant components of the

same tetrad can be obtained by using g^v to raise the index v, ea(i = g^veav. The labels on the vectors have no tensorial meaning, but for convenience we shall raise and lower them by using r|ab = (1,-1, -1,-1). We have

Quantizations of Scalar in Constant-Linear-Acceleration Frames

367

rn = Y a e / ,

eyVab-guv,

e^e^gnv^ab

e ^ = g^ v e a \

(23.42)

where y a are the usual constant Dirac matrices, and the Latin suffixes a, b, c,... may be called Lorentz indices because they are related to the Lorentz group. We have the following expressions for the non-zero diagonal elements: , = (W2,-1,-1,-1)

e a i l =(W, - 1 , - 1 , - 1 ) ,

T)ab = ( 1 , - 1 , - 1 , - 1 ) ,

e ^ = (W, 1, 1, 1),

e / = (W-!, 1, 1, 1),

e^MW"1,-1,-1,-1).

(23.43)

These vierbeins in (23.41) correspond to "scale gauge fields" (which differ from Yang-Mills fields or gauge fields) in the Poincare gauge-invariant involving fermions.

Lagrangian

4

One can introduce the local Lorentz transformation of the spinor ip -» exp[iEab(x)aab]iJ.,

a

ab=(i/2![Ub-¥al-

(23.44)

Corresponding to this local transformation, one has the gauge covariant partial derivative

The connection co^fx) is introduced so that one can gauge the Lorentz

group.

It is given by 5

cVb(x) = \ e*ao ii e*b _ ^ b )

+

I

e pae^(a x e p c

- apef

)eilC - ( a « b )

= - e ^ W 8 b ( \ 0 + e xb 3 x W5 a0 8^ 0 . There are only two non-vanishing components for co(iab(x):

368

Principle of Relativity in the Physics of Accelerated Frames

(23.46)

a) 0 oi = - a ) 0 1 0 = -fl 1 W.

(23.47)

It follows from (23.45) and (23.47) that

V(rt> = [d0 + 7 Y°Y ^iWhp,

VkiJ; = B^y,

for

(23.48)

k = 1,2,3.

One can verify that the results in (23.48) are precisely the same as V ^ , with V^ = (do +T( a k w )v°Y k . ai> s2' %)> i n (23.41) because the metric tensor g^v or the Wu factor W does not depend on y and z, i.e. d2W=d3W=0. From the Lagrangian (23.41), we obtain the generalized Dirac equation for CLA frames:

( i m ^ - m t o = 0,

^(-"^iri*- m) = 0.

(23.49)

For covariant quantization of the spinor fields in such CLA frames, we use the gauge covariant derivative (23.45) or (23.41) consistently and follow the canonical quantization as closely as possible.

The "canonical momentum" Jtb

conjugate to ^b is defined as

Jtb = dLq/d(V0yb) = h|)VW,

L^ = WTJJrW V^a|> - Wimin|>,

(23.50)

where the presence of W(w,x) in the Lagrangian, the canonical momentum (23.50) and the field equations (23.49) are the only source of complication of field quantization in CLA frames.

The Hamiltonian density H^ for a free

electron is defined as

Hy = jiv0i|) - L ^ - V ^ g ^ y k i a i ^ + V^g m w ,

Quantizations of Scalar in Constant-Linear-Acceleration Frames

(23.51)

369

where we have used r k , k=l,2,3, in (23.40) and Vk in (23.41). The energy-momentum tensor T^v for the spinor field is defined by

^"Wi™-1^-

(23 52)

-

Thus, we have the energy H and momentum P k of the spinor field: H= Jd 3 xT° = f d\-fg(-Hfiykdky+mw),

Pk= Jd3xT°= fd3x£^iw°Vky

T° =H^,

= fd*x±yhky.

(23.53)

(23.54)

The fundamental equal-time anticommutation relations for spinor fields are postulated to be {i|>a(w, x), jtp(w, y){ = i53(x - y)8ap, {ip a (w, X), t|)p(w, y)} = 0,

(23.55)

iJl a (w, X), Jtp(w, y)} = 0,

where Jt a (w,y) is given by (23.50) and W must be treated as an ordinary function rather than an operator. From equations (23.53) through (23.55), one can verify that i [ P ^ ( x ) ] = Vi.(x),

P^=(H,Pk).

(23.56)

In the calculation of [Pk, Hi(x)] , the extra function 1/W in (23.54) disappears after the anticommutator (23.55) is used. Similar cancellations of W's occur in [H, *i>(x)] when (23.55) and the free equation irkVkH> - mH> = - ir°Voi|) in (23.49) are used. In order to carry out quantization in momentum space in terms of the creation and annihilation operators, we must first find the solution to the generalized free Dirac equation (23.49) in linearly accelerated frames. Because

370

Principle of Relativity in the Physics of Accelerated Frames

of the presence of the gauge covariant derivative V^, the time and space dependence of the wave function t|> is not just the phase iP(x) for a free fermion.

There is an additional gauge factor G(w), which

changes

the

magnitude of the "plane wave function," due to the acceleration of reference frames: ip = u(k)exp[-iP(x) - G(w)],

(r*

r

P(x) = P(x^) = P(w,x,y,z),

m)u(k) = 0,

(23.57)

(23.58)

where u(k) is a 4-component spinor. The phase factor P(x) is given by (23.19) and the time-dependent gauge factor G(w) is found to be W

I

1

G(w)=/dw-Y 0 Y 1 (3iW) = iY 0 Y 1 tanh- 1 (a o w + p o ),

for

W = Y2[Yo"2+a0x], (23.59)

G(w) = I / V a ' w ,

for

w h e r e Y=l/^l-(a 0 w+p 0 ) 2 , Yo=l/\/l~Po2>

W = [l+Yoa*x] :

an

d Y°Y1 satisfies the relation (Y°Y 1 ) 2 =1

because of the anti-commutation in (23.40). For a free spinor field tp, the Fourier expansion of a solution (23.57) for a "free" electron takes the form

Im *|)(w,x) = f,—

k

d3k , 3

2

j

^ -t) s=l

[b(k,s)u(k,s)exp(-iP - G)

+ dt(k,s)v(k,s)exp(iP - G)],

k0=W(ok,

(23.60)

cat =Vk 2 +m 2 ,

where the phase P=P(x) is given by (23.19) and the gauge factor G=G(w) is given by (23.58) and (23.59). According to (23.58), the spinors u(k,s) and v(k,s)

Quantizations of Scalar in Constant-Linear-Acceleration Frames

371

satisfy the equations ( r ^ k r m)u(k,s) = 0,

iifk.sMr^- m) = 0, (23.61)

(rvk^ m)v(k,s) = 0,

i

v(k,s)(n k (l + m) = 0,

and the orthonormality relations u(k,s)u(k,s')= -v(k,s)v(k,s')= 8SS',

ut(k,s)u(k,s')= vt(k,s)v(k,s')= ^ - S s s ' ,

k0=wVk2+m2.

(23.62)

v(k,s)u(k,s') = u(k,s)v(k,s') = 0. We normalize the spinors so that ut(k,s)u(k,s') =(ko/m)5SS'. This is a covariant relation since both sides transform like the zeroth component of a 4-vector under accelerated Wu (or MWL) transformations. We also have the following completeness relations

2) [ua(k,s)up(k,s) - va(k,s)vp(k,s)] = 5ap, s=l

2 2)

_ /m + r^k^\ ua(k,s)up(k,s) = ^ ^ J a p,

• 2 va(k,s)vp(k,s) - ^

^

(23.63)

J a p.

Upon second quantization of the spinor field, the Fourier expansion coefficients

372

b(k,s), bt(k',s'), d(k,s) and dt(k',s') become operators that

Principle of Relativity in the Physics of Accelerated Frames

annihilate and create particles. anticommutation relations,

They are assumed to satisfy the

ib(k,s), bt(k,,s')}-63(k-k,)»ss-> (23.64) jd(k,s), dt(k',s')S = 5 3 ( k - k ' ) 8 s s . , where all other anticommutators vanish.

The anticommutators for the

quantized fields ij>(w,x) and i|>t(w,x) in (23.55) can be shown to be consistent with (23.60) and (23.64). In the calculation of these anticommutators, the gauge factor G(w) in y and n>t cancels because of the relation Y°Y1 = -Y 1 Y°, or

e-G(w)Y0=YOeG(w);

G(w) = /dw-Y°Y 1 (3iW).

(23.65)

Finally, the Hamiltonian operator H=/d 3 xT° 0 cannot be expressed i n terms of the operators b, bt, d and dt because of the presence of W in the integrand.

This restriction occurs because the Hamiltonian H transforms like

k 0 and ko=W(% =wVk 2 +m 2 . However, we have Hm = / d 3 x T ° / W = / d 3 k w k [ b t ( k ) b ( k ) + d(k)dt(k)],

(23.66)

Pi = J d 3 x T ° = / d 3 k k i [bt(k)b(k) + d(k)dt(k)].

(23.67)

The quantity Hm in (23.66) is not exactly the Hamiltonian, in contrast to H i n (23.53). Instead, it corresponds to po/W which is useful for treating scattering processes in field theory. 23 d.

Quantization of the electromagnetic field in CLA frames The action of the electromagnetic field is assumed to be that of the usual

invariant form with a gauge fixing term involving a gauge parameter p,

Quantizations of Scalar in Constant-Linear-Acceleration Frames

373

S=/d4xLem,

Lem—^V^ f ^

v

-^g(D%)2,

fM.v = D (i a v -D v a M . = a^av - ava^,

(23.68)

(23.69)

I>a|1-D|1ai'-Dll(aagi»)-all(aagi«')-aagi«I-^all>rg V-g

= g n % a „ - a„ - p = a^( gn«V^g ) ^ x. V-g

(23.70)

Varying the action S, we obtain the free electromagnetic equation

Since f^v is antisymmetric, the scalar field X^DH-a^ satisfies the free equation i n CLA frames

- j L av( giW^g a^x) = DTO^x =0, V-g

(23.72)

which corresponds to the free equation aiia(ix=0 in inertial frames. The electromagnetic 4-potential can be written in the form

/

I

d^k

3 r 2 [a(k,a)e^(a)exp(-iP(x))

V(2jt)32cok a=0

+ at(k ) a)e ( 1 (a)exp(iP(x))], ^, ^

E[i (a)E v (a)

k 0 = Wcok = Wlkl.

k k = -g(iv+(l-p)-^,

^

374

Principle of Relativity in the Physics of Accelerated Frames

(23.73) (23.74)

[a(k,a), at(k',a')] = - 53(k-k') g aa ', [a(k,a), a(k',a')] = 0,

(23.75)

[at(k,a), at(k',a')] = 0.

The sum over a in (23.73) for four linearly independent unit polarization vectors e^a), a=0,1,2,3 is an ordinary sum and not a scalar product in 4-dimensional space. It can be quantized covariantly using the Gupta-Bleuler method generalized for non-inertial frames.6 Equation (23.71) is important for covariant quantization of the electromagnetic field by the Gupta-Bleuler method which involves both physical photons (i.e., the two transverse components) and unphysical photons (i.e., longitudinal and time-like photons). In the Gupta-Bleuler method, equation (23.71) guarantees that the unphysical photons do not interact and hence, do not contribute to physical amplitudes and can be excluded from physical states. 7 In other words, it assures that when the condition for physical states Iw), 3

k^atl(k)l